What is the basis for King Melchizedek actually being Jesus? Where did this idea come from?

Melchizedek is mentioned in Genesis, Psalms, and Hebrews.

Melchizedek is introduced shortly after Abram rescued his nephew Lot:

Then Melchizedek king of Salem brought out bread and wine; he was the priest of God Most High.
And he blessed him and said:
| Blessed be Abram of God Most High,
| Possessor of heaven and earth;
| And blessed be God Most High,
| Who has delivered your enemies into your hand.
And he gave him a tithe of all.
— Genesis 14:18–20

The bread and wine is considered a foreshadowing of the symbols of Christ's sacrifice.

Later, in Psalms 110, King David mentions Melchizedek:

…
The LORD said to my Lord,
| Sit at My right hand,
| Till I make Your enemies Your footstool.
…
The LORD has sworn
And will not relent,
| You are a priest forever
| According to the order of Melchizedek.
The Lord is at Your right hand;
He shall execute kings in the day of His wrath.
He shall judge among the nations,
He shall fill the places with dead bodies,
He shall execute the heads of many countries.
…

The latter part obviously foreshadows the events described in Revelation when Christ returns.

But more significantly, notice "The LORD said to my Lord", foreshadowing the knowledge of both Father and Son, and "Sit at My right hand", which describes the current position of the Son with respect to the Father.

Clearly David is writing about the being that later became Jesus.

Now notice that Jesus is called "a priest forever". Jesus always was and always will be a priest to mankind.

In Hebrews, Paul confirms this by calling Jesus "High Priest forever according to the order of Melchizedek":

For this Melchizedek, king of Salem, priest of the Most High God, who met Abraham returning from the slaughter of the kings and blessed him, to whom also Abraham gave a tenth part of all, first being translated “king of righteousness,” and then also king of Salem, meaning “king of peace,” without father, without mother, without genealogy, having neither beginning of days nor end of life, but made like the Son of God, remains a priest continually.
— Hebrews 7:1–3

Without being blasphemous, who other than Jesus could be called "King of Peace" or "King of Righteousness"? Who other than Christ can hold the titles of both King and Priest? Who else could be described as "having neither beginning of days nor end of life, but made like the Son of God"?

And most significantly, who else but Jesus could it be that "remains a priest continually"?

But He, because He continues forever, has an unchangeable priesthood. Therefore He is also able to save to the uttermost those who come to God through Him, since He always lives to make intercession for them. For such a High Priest was fitting for us, who is holy, harmless, undefiled, separate from sinners, and has become higher than the heavens; who does not need daily, as those high priests, to offer up sacrifices, first for His own sins and then for the people’s, for this He did once for all when He offered up Himself. For the law appoints as high priests men who have weakness, but the word of the oath, which came after the law, appoints the Son who has been perfected forever.
— Hebrews 7:24–28

Clearly Melchizedek and the immortal being that was later incarnated as Jesus and raised as Christ and Saviour were the same person.

Much of the above was inspired by pages 24 and 25 of Who Is God? | United Church of God.

The chief basis for the belief is Hebrews 7:3, which says in the NET translation:

Without father, without mother, without genealogy, he has neither beginning of days nor end of life but is like the son of God, and he remains a priest for all time.

Those who argue that he is Jesus say that a plain reading of the verse is that Melchizedek literally had no parents or ancestors, and that he never died and is eternally a priest.

Others would counter it by saying that he was a type of Christ, and that Hebrews is calling attention to his mysterious, spontaneous appearance in Genesis: no genealogical record, no record of his life beyond his interaction with Abraham. They also point out that Hebrews says he was made "like" the Son of God, not that he was the Son of God, which is a title that Hebrews reserves for Jesus.

As supporting evidence, proponents of the "Melchizedek = Jesus" view point out that only a few chapters after Abraham's encounter with Melchizedek, he meets someone who has traditionally been identified as a manifestation of God. So such encounters were not unheard of.

The origin of the belief is murky, but it seems to have been a minority opinion in relatively early times, and remains so today. It seems to not have a lineage per se, but rather that commentators and theologians arrive at the conclusion independently at different times. Its earliest mention that I could find is the fifth century.

According to Philip Edgecomb Hughes' commentary on Hebrews, Jerome asserted (in letter 73, to Evangelus) that Origen believed Melchizedek to be an angel and that he himself along with Irenaeus, Hippolytus, Eusebius of Caesarea, Apollinaris, and Eustathius believed him to be a human being and a type of Christ. Hughes adds Cyril of Alexandria and Epiphanius to the list. But Ambrose (according to Hughes) sometimes referred to Melchizedek as God, sometimes the Son of God, and sometimes a type of Christ. Epiphanius was an early writer who explicitly wrote against the idea of Melchizedek being a theophany; John Brown also mentioned that he was aware of the belief. Hughes identifies Cunaeus as a more recent proponent of the belief, along with J.B. McCaul and A.T. Hanson, about whom not much information is out there online. In addition to all this (just as a side note) there were various heretical sects that saw Melchizedek as a manifestation of the Holy Spirit, and Jewish writers commonly identify Melchizedek with Shem.