Paul's letters are the oldest parts of the NT corpus so it is a shock to realize how little they refer to Jesus and later material found in the gospels. By the same token we may ask the question in reverse: what do the gospels know of Paul?
Let's take the first question: what does Paul know about Jesus? Paul knows that Jesus descended from David "according to the flesh" (Romans 1:3) so Paul knows that Jesus was Jewish. From the same passage, Paul knows about Jesus' resurrection. Paul's gospel of God describes God's designation of Jesus as Son "in power, according to the spirit of holiness" by his resurrection from the dead (Romans 1:4). As we know, Paul has an encounter with the resurrected Jesus. He describes it in Gal 1: "God was pleased to reveal his Son to (or "in") me so that I might preach him among the Gentiles (1:16).
Paul hands on to the Corinthians traditions about Christ that he has received: that Christ died for our sins according to the scriptures, that he was buried,that he was raised on the third day in accordance with the scriptures, that he appeared to Cephas and the 12, to five hundred brethren, to James, to all the apostles, and finally to him (I Cor 15:3-9). This creedal declaration uses active and passive verbs easily committed to memory: Christ died, was buried, was raised, appeared.
Paul knows that there are male and female apostles before him some of whom are in Jerusalem (Gal 1:17) and some known more widely (Rom 16:7). He knows James, the Lord's brother (Gal 1:19). He knows Cephas (Peter) well enough to stay with him for 15 days (Gal 1:18). He knows John, another of the Jerusalem leaders (Gal 2:9).But he is not known by sight to "the Judean communities in Christ." (Gal 1:22). They have only heard of him as a former persecutor who is now preaching the faith he formerly tried to destroy.
Paul knows traditions around the passion narrative beginning from the night of Jesus' arrest. He has received these traditions about eating bread and drinking the cup and he hands them on to the Corinthians: "The Lord Jesus, on the night that he was betrayed, took bread, and when he had given thanks, he broke it" (I Cor 11:24-25).
Paul preaches to communities first. His letters clarify earlier preaching. What he preaches is the gospel of God (I Thess 2:2, 8-9; 2 Cor 11:7), the gospel (Rom 1:16), or the gospel of Christ (Rom 15:19; I Cor 9:12; 2 Cor 12:12). Thus, the noun "gospel" is a technical term for the message of Christ and its proclamation. The content of the gospel can be explicated in different ways as we see in Paul's letters.
Similar language appears in the gospels. In Mark 1:14, Jesus proclaims the gospel of God. Mark then describes Jesus saying, "The time is fulfilled, the kingdom of God is at hand, repent and believe in the gospel" (Mark 1:15). But what is the content of this gospel? The content of Mark's gospel is Jesus' death and resurrection (Mark 8:35). When Jesus' announces the gospel (of God) in Mark 1:14-15, it is as yet unknown but it is given content by the narrative of the gospel. Mark 1:1 gives content to the gospel as that of Jesus Christ (some manuscripts add: Son of God) as it is written in Isaiah the prophet. Those who titled the gospels, "The Gospel according to Matthew/Mark/Luke" correctly discerned the narrative meaning of gospel explicated in Jesus' life, death and resurrection.
So what the synoptic gospels did was to expand (I could choose another verb here: alter? correct?) the meaning of Paul's gospel to include the narrative of Jesus' life through the prism of his death and resurrection. Mark's emphasis is on Jesus' death. (Much more could be said about the different Christologies of the gospels and Paul).
When I speak of the gospels in relation to Paul's letters, I am trying to take all the canonical evidence of the New Testament seriously. I am asking a question from within a faith tradition. From an historical perspective, its hard to say that the writers of the synoptic gospels were in dialogue with Pauline communities because we have hardly any evidence that they were. But at least in use of the term "gospel" we can see an overlap in terminology and content between Paul's proclamation of Jesus' death and resurrection and Mark's understanding of "gospel."