Question: Where is the Pope in the Bible?
Answer: This question presupposes that everything we believe as Christians has to be found in the Bible....which is something that the Bible never declares.
Our faith/doctrines/teachings don't come from a book, no matter how holy.
Our faith comes from Christ, through His Apostles, to the Church. Catholicism was whole and entire* before a single word of the New Testament was ever put to writ. The Bible reflects what was already proclaimed by the Apostles and their successors, but is not the source of our faith.
And this Sunday's readings do indeed reflect the teachings that Jesus founded His Church on Peter and Peter was given authority over the entirety of the Body of Christ. That is, this Sunday's readings reflect the concept of the papacy.
In the first reading we see the keys to the palace being given to Eliakim, who is the king's prime minister:
I will place the key of the House of David on Eliakim's shoulder;when he opens, no one shall shutwhen he shuts, no one shall open.--Isaiah 22
Being given the keys was a hallmark of authority. When the king is away, Eliakim is in charge.
And in the Gospel we see the keys to the kingdom being given to Peter:
I will give you the keys to the kingdom of heaven.Whatever you bind on earth shall be bound in heaven;--Matt 16
When the King is away, Peter is in charge.
Thus, what Peter "binds", that is, declares authoritatively that his flock is obligated to do, heaven "binds" as well. And what Peter "looses", is "loosed" in heaven. It is a reflection of the authority of Peter to teach infallibly on matters of faith and morals.
From Scott Hahn:
"When Jesus gives to Peter the keys of the kingdom, Peter is receiving the Prime Minister's office, which means dynastic authority from the Son of David, Jesus, the King of Israel, but also an office where there will be dynastic succession. When I discovered that, it was like the blinders fell off. Within a few weeks I had gotten together with the leading Protestant theologians in the world, one of the most reputable anti- Catholic Protestant theologians and spent ten hours with him and then in a Mercedes we drove two hours and I presented this case, and his only comment was, "That's clever." But he said, "You don't have to follow the Pope because of that." I said, "Why not?" And he said, "Well, I'm going to have to think about it." He said, "I've never heard that argument before." And I said, "It' s one of the basic arguments that Cajeton used against the Protestants in the 16th Century and Cajeton was one of the most well-known defenders of the Catholic faith and you've never heard of him before?" I said, "I had never heard of it before until I discovered it on my own and then found it in all these other people." And he said, "That's clever." Clever, perhaps. True, definitely; enlightening, illuminating, very interesting....The role of Peter as steward of the kingdom is further explained as being the exercise of administrative authority as was the case of the Old Testament chamberlain who held the keys."Now, what he means there is that nowhere else, when other Apostles are exercising Church authority are the keys ever mentioned. In Matthew 18, the Apostles get the power to bind and loose, like Peter got in Matthew 16, but with absolutely no mention of the keys. That fits perfectly into this model because in the king's cabinet, all the ministers can bind and loose, but the Prime Minister who holds the keys can bind what they have loosed or loose what they have bound. He has, in a sense, the final say. He has, in himself, the authority of the court of final appeal and even Protestants can see this.In fact, I found this quotation in Martin Luther from 1530, years after he had left the Church, "Why are you searching heavenward in search of my keys? Do you not understand, Jesus said, 'I gave them to Peter. They are indeed the keys of heaven, but they are not found in heaven for I left them on earth.'" This is Jesus talking, "'Peter's mouth is my mouth, his tongue is my key case, his keys are my keys. They are an office.'" Luther even saw it, "'They are a power, a command given by God through Christ to all of Christendom for the retaining and remitting of the sins of men.'" The only thing that Luther won't admit is that there was succession after Peter died, which is exactly what the keys denote, given their Old Testament." Dr Scott Hahn on the Papacy | Catholic-Pages.com
Until the King returns, the one in authority is holder of the Keys!
*Although Catholicism was whole and entire from the moment the Apostles received the Holy Spirit at Pentecost, it is true that our understanding of this kerygma developed. Doctrinal development, increased clarity of the teachings, further explication of what has been revealed is ongoing.