I will come back to add more references, but the topic of faith and science comes up in many of Bishop Robert Barron videos, talks and posts. From this paragraph, it seems that the answer is in some sense in support of the non-overlapping magesteria:

The obvious success of the physical sciences—evident in the technology that surrounds us and facilitates our lives in so many ways—has convinced many of our young people (the vast majority of those who watch YouTube are young) that anything outside of the range of the empirical and measurable is simply a fantasy, the stuff of superstition and primitive belief. That there might be a dimension of reality knowable in a non-scientific but still rational manner never occurs to them. This prejudice, this blindness to literature, philosophy, metaphysics, mysticism, and religion is the scientism that I’m complaining about.

But in another sense, it is not that there are two different versions of truth (the scientific version and the religious version), or even that the two point to the same truth but in isolation of each other. Science and faith depend upon each other:

the modern physical sciences were, in fact, made possible by the religious milieu out of which they emerged. It is no accident that modern science first appeared precisely in Christian Europe, where a doctrine of creation held sway. To hold that the world is created is to accept, simultaneously, the two assumptions required for science, namely, that the universe is not divine and that it is marked, through and through, by intelligibility.

That is, science does not make sense without the philosophical underpinnings that the Christian faith provides.

Bishop Barron frequently cites Popes Benedict and St. John Paul as a source of these arguments.

There are claims made by Hugh Ross and others that the Catholic Church endorses the "non-overlapping magesteria" model for the relationship between science and religion. A problem with these claims is that the Church rarely ties itself down to a non-negotiable position as clearly as this. The result is that these specific claims are liable to be interpretations of what are often intended as ambiguous statements. As well as Ross, we find Wikipedia says the Church believes in Non-Overlapping Magisteria, but that where science conflicts with the Roman Catholic faith position, it gets confusing. RationalWiki is more cautious, saying that Non-Overlapping Magisteria may have helped reconcile the Roman Catholic Church to evolution though this is disputed.

To my knowledge, Pope Francis does not use the term "non-overlapping magisteria", but in his Encyclical Letter, LAUDATO SI, he makes a quite clear statement that the Church does not presume to settle scientific questions. He goes on to indicate that the limits of his authority on matters of science were to encourage honest and open debate for the common good:

Laudato Si 188: There are certain environmental issues where it is not easy to achieve a broad consensus. Here I would state once more that the Church does not presume to settle scientific questions or to replace politics. But I am concerned to encourage an honest and open debate so that particular interests or ideologies will not prejudice the common good.

According to Laudato Si, it would therefore appear that the current Catholic understanding of the relationship between science and religion is that science determines matters of fact in the natural world, and the Church concerns itself with the moral issues that arise out of science. Forgetting the semantics, this is what Stephen J. Gould meant by "Non-Overlapping Magisteria".

No, the Church does not have a separatist attitude toward science.

There have been periods in which, admittedly, some persons in the hierarchy of the Church have viewed the natural sciences with a certain suspicion (one thinks of the condemnation of Galileo), but such an attitude has neither been the norm nor is it compatible with Church teaching.

Magisterial Documents

The most important recent Magisterial document on the relationship between faith and reason in general (including its relationship to science) is John Paul II’s Fides et ratio (FR).

The main thrust of that encyclical is that there can never be any fundamental conflict between the various uses of human reason and the contributions of faith. As the very opening words of the encyclical say,

Faith and reason are like two wings on which the human spirit rises to the contemplation of truth; and God has placed in the human heart a desire to know the truth—in a word, to know himself—so that, by knowing and loving God, men and women may also come to the fullness of truth about themselves (FR, prologue).

The fundamental reason for this is that all truth, regardless of its immediate origin, has its ultimate origin in God, who is the Truth Itself. Jesus, argues the encyclical, even declares that he is the “way, and the truth, and the life (Jn 14:6; see FR 2).

Hence what reason discovers in one field—including the sciences—cannot truly contradict what it knows through faith.

(It may interest the reader that traditionally both sciences and faith are considered habits of the intellect, that permit them to know certain things. Sciences are acquired by study, whereas faith is infused by grace. However they both modify the same faculty: the intellect. Have a look, for instance, at the Summa theologiae, I-IIae, q. 54, a. 1, responsum, where he gives the various sciences as examples of habits of the intellect; and q. 62, a. 3, in which he defines faith as a theological virtue—a “virtue” being a good habit, as opposed to a “vice”.)

A similar idea can be found in the Catechism of the Catholic Church, especially No. 159, which quotes both Vatican I’s dogmatic constitution Dei Filius (DF) and Vatican II’s pastoral constitution Gaudium et spes (GS):

“Though faith is above reason, there can never be any real discrepancy between faith and reason. Since the same God who reveals mysteries and infuses faith has bestowed the light of reason on the human mind, God cannot deny himself, nor can truth ever contradict truth” (DF 4).

“Consequently, methodical research in all branches of knowledge, provided it is carried out in a truly scientific manner and does not override moral laws, can never conflict with the faith, because the things of the world and the things of faith derive from the same God. The humble and persevering investigator of the secrets of nature is being led, as it were, by the hand of God in spite of himself, for it is God, the conserver of all things, who made them what they are” (GS 36).

Other evidence of the Church’s non-hostility to science

It is telling that these sources come from very different periods in Church history: Vatican I, at a time when the Church felt threatened by the modern world (as evidenced by Pope Pius X’s Pascendi dominici gregis quoted by Geremia, written 37 years after Vatican I), and Vatican II (as well as John Paul II’s encyclical), at a time when the Church felt called to engage the modern world. Yet they teach essentially the same idea: that all truth participates in the One Truth, who is God.

The Church has never claimed competence in scientific matters (as Dick Hartfield correctly mentions). Indeed the competency of the Magisterium as such is only in faith in morals:

And this infallibility [i.e, the infallibility of the whole Church] with which the Divine Redeemer willed His Church to be endowed in defining doctrine of faith and morals, extends as far as the deposit of Revelation extends, which must be religiously guarded and faithfully expounded (Vatican II, Lumen gentium, 25, among other references).

It is also telling that the Church has been a patron of the sciences for a long time. Many distinguished scientists in history were Catholics, among them Nicolas Copernicus, Galileo Galilei (despite his run-in with the Inquisition, he remained a Catholic to the end of his days), René Descartes, Gregor Mendel, and Louis Pasteur. What is perhaps more interesting is that many of these were clergy: Copernicus and Mendel being probably the most famous, but there are many others. To this day, the Vatican maintains an observatory in Castel Gandolfo, Italy, as well as the Pontifical Academy of Sciences.

Conclusion

In summary, the Church has historically taken (and continues to take) the position that there is no fundamental conflict between faith and science. Although there have been individual episodes (e.g., Galileo) in which particular scientists or particular scientific ideas have met with resistance by the Church’s hierarchy, the Church’s teaching has always been that faith and science can never truly conflict, since they both participate in the very same source of truth: namely, God.

(It is worth noting, for instance, that the Inquisition did not persecute Nicolas Copernicus for his heliocentric model; it was not considered a matter of faith or morals. What got the Inquisition’s attention with Galileo was that he proceeded to try to interpret the Scriptures based on those findings, something that the Inquisition regarded as a task for theologians—doubtless guarding their privilege too jealously. See the very well written Wikipedia article.)

The "'non-overlapping magesteria' model for the relationship between science and religion" is not Catholic. In fact, it is condemned as a part of the heresy of Modernism, what Pope St. Pius X called the "synthesis of all heresies" in his 1907 encyclical condemning Modernism, Pascendi Dominici gregis (my emphases):

Faith and Science

16. Having reached this point, Venerable Brethren, we have sufficient material in hand to enable us to see the relations which Modernists establish between faith and science, including history also under the name of science. And in the first place it is to be held [according to the Modernists] that the object of the one is quite extraneous to and separate from the object of the other. For faith [according to the Modernists] occupies itself solely with something which science declares to be unknowable for it. Hence [according to the Modernists] each has a separate field assigned to it: science is entirely concerned with the reality of phenomena, into which faith does not enter at all; faith on the contrary concerns itself with the divine reality which is entirely unknown to science. Thus the conclusion is reached that there can never be any dissension between faith and science, for if each keeps on its own ground they can never meet and therefore never be in contradiction. And if it be objected that in the visible world there are some things which appertain to faith, such as the human life of Christ, the Modernists reply by denying this. For though such things come within the category of phenomena, still in as far as they are lived by faith and in the way already described have been by faith transfigured and disfigured, they have been removed from the world of sense and translated to become material for the divine. Hence should it be further asked whether Christ has wrought real miracles, and made real prophecies, whether He rose truly from the dead and ascended into heaven, the answer of agnostic science will be in the negative and the answer of faith in the affirmative—yet there will not be, on that account, any conflict between them [according to the Modernists]. For it will be denied by the philosopher as philosopher, speaking to philosophers and considering Christ only in His historical reality; and it will be affirmed by the speaker, speaking to believers and considering the life of Christ as lived again by the faith and in the faith.

Faith Subject to Science

17. Yet, it would be a great mistake to suppose that, given these theories, one is authorised to believe that faith and science are independent of one another. On the side of science [according to the Modernists] the independence is indeed complete, but it is quite different with regard to faith, which is subject to science not on one but on three grounds [according to the Modernists:]

1. For in the first place it must be observed that in every religious fact, when you take away the divine reality and the experience of it which the believer possesses, everything else, and especially the religious formulas of it, belongs to the sphere of phenomena and therefore falls under the control of science. Let the believer leave the world if he will, but so long as he remains in it he must continue, whether he like it or not, to be subject to the laws, the observation, the judgments of science and of history.
2. Further, when it is said that God is the object of faith alone, the statement refers only to the divine reality not to the idea of God. The latter also is subject to science which while it philosophises in what is called the logical order soars also to the absolute and the ideal. It is therefore the right of philosophy and of science to form conclusions concerning the idea of God, to direct it in its evolution and to purify it of any extraneous elements which may become confused with it.
3. Finally, man does not suffer a dualism to exist in him, and the believer therefore feels within him an impelling need so to harmonise faith with science, that it may never oppose the general conception which science sets forth concerning the universe.

Thus it is evident [to the Modernists] that science is to be entirely independent of faith, while on the other hand, and notwithstanding that they are supposed to be strangers to each other, faith is made subject to science. All this, Venerable Brothers, is in formal opposition with the teachings of Our Predecessor, Pius IX, where he lays it down that: In matters of religion it is the duty of philosophy not to command but to serve, but not to prescribe what is to be believed but to embrace what is to be believed with reasonable obedience, not to scrutinise the depths of the mysteries of God but to venerate them devoutly and humbly.

The Modernists completely invert the parts, and to them may be applied the words of another Predecessor of Ours, Gregory IX., addressed to some theologians of his time: Some among you, inflated like bladders with the spirit of vanity strive by profane novelties to cross the boundaries fixed by the Fathers, twisting the sense of the heavenly pages…to the philosophical teaching of the rationalists, not for the profit of their hearer but to make a show of science…these, seduced by strange and eccentric doctrines, make the head of the tail and force the queen to serve the servant.

Pope Pius XII's 1950 encyclical Humani Generis was primarily about re-condemning Modernism in the form of the heretical "New Theology" that had sowed its rotten seed in seminaries and theological schools since Pope St. Pius X's time.