What is the biblical basis for Jesus being able to redeem the Jewish people in his second coming?

According to the NT writings the whole World has been redeemed by the death of Jesus, including Israel. The act is finished. This understanding about the redemption of all was explained by Origen c. 250.

Huffington Post May 23, 2013 "Pope Francis rocked some religious and atheist minds today (May 23, 2013) when he declared that everyone was redeemed through Jesus, including atheists."

Redemption is, of course, not the same as salvation.

Okay here is some additional explanation - it is only a quick answer. I ought to spend a week thinking it out properly.

In the NT, the Greek word for redemption means a releasing effected by payment of ransom 1a) redemption, deliverance 1b) liberation procured by the payment of a ransom

It is the price that the Redeemer paid for our release from the bondage of sin. With this payment the virtues that God withheld from the children of Adam, are now offered to all mankind. We have but to ask sincerely and deeply to receive these virtues. The two virtues are summarized as "holiness and justice." As Origen has explained, the absence of these virtues in the children of Adam, and his descendants, was like a stain on mankind. This stain is washed away by the incarnation and sacrifice of Jesus - we are "washed clean" by His blood. Redemption for us then, is a washing away of the stains that were "original sin" - a term used by the Church Fathers - St. Augustine and St. Thomas. Origen before them, and others Fathers.

This is my understanding, I am sure that a Priest or theologian could explain it better.

Salvation, on the other hand begins with a life centered on God, trusting in God, and loving God. Someone must show us how to repent, for this is to love God with all our heart mind body and soul. The Hebrew practice is Teshuva, a lifetime of improving our aim so we get closer to the mark - as Paul taught.

Jesus said: "be ye perfect, as your Father in Heaven is perfect."

Paul showed his followers how to realize The Christ within them. Jesus said to the twelve "the kingdom of God is within you."

Simply put: redemption is the price paid for the removal of the stains on us that had, until the crucifixion, prevented us from re-entering Paradise. Salvation is the process by which God makes us worthy to enter Paradise. (As the NT puts it: We must have on our wedding garments provided by God to enter the bridal chamber.)

This is a simple explanation of very complex theology, and you should do the research yourselves. My explanation is taken from Anglican Doctrine.

The idea is not that all Jews will be reconciled to God, but that a remnant will repent and return to the Lord in the last days. Paul explains it this way:

(speaking of Israel:) I say then, God has not rejected His people, has He? May it never be! For I too am an Israelite, a descendant of Abraham, of the tribe of Benjamin. God has not rejected His people whom He foreknew. Or do you not know what the Scripture says in the passage about Elijah, how he pleads with God against Israel? . . . But what is the divine response to him? “I have kept for Myself seven thousand men . . . .” In the same way then, there has also come to be at the present time a remnant according to God’s gracious choice. . . .

(speaking of Israel:) I say then, they did not stumble so as to fall, did they? May it never be! But by their transgression salvation has come to the Gentiles, to make them jealous. Now if their transgression is riches for the world and their failure is riches for the Gentiles, how much more will their fulfillment be! . . . For if their rejection is the reconciliation of the world, what will their acceptance be but life from the dead? . . .

(speaking of Israel as an olive tree, and the people as its branches:) But if some of the branches were broken off, and you, being a wild olive, were grafted in among them and became partaker with them . . . do not be arrogant toward the branches . . . . they also, if they do not continue in their unbelief, will be grafted [back] in, for God is able to graft them in again. For if you were cut off from what is by nature a wild olive tree, and were grafted contrary to nature into a cultivated olive tree, how much more will these who are the natural branches be grafted into their own olive tree?

For I do not want you, brethren, to be uninformed of this mystery—so that you will not be wise in your own estimation—that a partial hardening has happened to Israel until the fullness of the Gentiles has come in; and so all Israel will be saved; just as it is written,

“The Deliverer will come from Zion, He will remove ungodliness from Jacob.” “This is My covenant with them, When I take away their sins.”

From the standpoint of the gospel they are enemies for your sake, but from the standpoint of God’s choice they are beloved for the sake of the fathers; for the gifts and the calling of God are irrevocable.

In summary:

• God has not rejected those Israelites whom He foreknew, but has kept a remnant for Himself

• Mercy came to the Gentiles in order to make the Israelites jealous

• Just as Israel's transgression resulted in mercy for the Gentiles, their reconciliation to God will result in life from the dead

• If Israel does not continue in their unbelief, God will welcome them back

• The Age of the Gentiles is temporary, and once the fullness of the Gentiles has come in, the partial hardening of Israel will be over

• God will do all of this out of His faithfulness to the promises He made to the fathers