The short answer is "yes", there are documents dating back to the early Church that suggest that the Bishop of Rome had a degree of authority superior to other bishops but it's not as straightforward as a clear consensus around our modern version of the papacy.

The best example of what I'm talking about would be the First Epistle of Clement. This was a letter composed probably in the first century AD by then Bishop of Rome Clement I and was sent to the people of Corinth. Clement would have been a contemporary of Saints Peter and Paul, traditionally he is held to have been consecrated as a bishop by St Peter himself and appointed as a successor.

The letter itself concerns a dispute in which the local bishop in Corinth had been deposed by the local Christian community. In the epistle, Clements instructs the Corinthians that they are to reinstate the deposed bishop. In the letter Clement gives a very early account of what Catholics now call "Apostolic Succession", the doctrine that the Twelve Apostles were the first bishops and that they appointed their successors. He makes it clear that laymen have no right removing their bishops, analogizing the Church to an army with common soldiers, generals, and a king at its head:

Let us then, men and brethren, with all energy act the part of soldiers, in accordance with His holy commandments. Let us consider those who serve under our generals, with what order, obedience, and submissiveness they perform the things which are commanded them. All are not prefects, nor commanders of a thousand, nor of a hundred, nor of fifty, nor the like, but each one in his own rank performs the things commanded by the king and the generals. The great cannot subsist without the small, nor the small without the great. There is a kind of mixture in all things, and thence arises mutual advantage. Let us take our body for an example. The head is nothing without the feet, and the feet are nothing without the head; yea, the very smallest members of our body are necessary and useful to the whole body. But all work harmoniously together, and are under one common rule for the preservation of the whole body.

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The apostles have preached the Gospel to us from the Lord Jesus Christ; Jesus Christ [has done sol from God. Christ therefore was sent forth by God, and the apostles by Christ. Both these appointments, then, were made in an orderly way, according to the will of God. Having therefore received their orders, and being fully assured by the resurrection of our Lord Jesus Christ, and established in the word of God, with full assurance of the Holy Ghost, they went forth proclaiming that the kingdom of God was at hand. And thus preaching through countries and cities, they appointed the first-fruits [of their labours], having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons. For thus says the Scripture a certain place, "I will appoint their bishops s in righteousness, and their deacons in faith."

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Our apostles also knew, through our Lord Jesus Christ, and there would be strife on account of the office of the episcopate. For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those [ministers] already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry. We are of opinion, therefore, that those appointed by them, or afterwards by other eminent men, with the consent of the whole Church, and who have blame-lessly served the flock of Christ in a humble, peaceable, and disinterested spirit, and have for a long time possessed the good opinion of all, cannot be justly dismissed from the ministry. For our sin will not be small, if we eject from the episcopate those who have blamelessly and holily fulfilled its duties. Blessed are those presbyters who, having finished their course before now, have obtained a fruitful and perfect departure [from this world]; for they have no fear lest any one deprive them of the place now appointed them. But we see that you have removed some men of excellent behaviour from the ministry, which they fulfilled blamelessly and with honour.

Presumably, the people of Corinth took Clement's admonition to heart because the First Epistle of Clement treated it as one of the canonical books of the bible for many years.

Now, the Epistle does not directly deal with the supremacy of the papacy, but it does do a few things. It establishes the concept of a Church lead by local bishops who derive their authority from having been appointed as successors to the Apostles by the same Apostles. It also establishes that the Bishop of Rome, was someone who the people of Corinth would have listened to even when they were rebelling against their own bishop. So, it's not a "slam dunk" demonstration of papal supremacy in the first century, but it does suggest that at least some of the structure of the Church was already in place shortly after the time of the Apostles.