The Idea in Brief
After his conversion and through the ministrations of Barnabas, Paul (then Saul) had met "the apostles" in Jerusalem (to include Cephas for fifteen days) who had provided him first-hand accounts of the life and teachings of Jesus. Later, Paul traveled on his missionary journeys with Barnabas, John Mark and Luke, who also had access to information concerning the life and teachings of Jesus (cf. Gospel of Mark and Gospel of Luke). However, there is one exclusive example in Paul's writing that is parallel to the Gospel of Matthew, and therefore suggests direct contact by Paul with Matthew and/or his writings. Thus Paul's sources of information concerning the life and sayings of Jesus appear to have been various; that is, with direct personal contact with living people who knew Jesus, Paul had access to information beyond the scope of what we have written in the four Gospel accounts.
Paul and Luke had spent extensive time together (during Paul's second missionary journey), and of course Luke authored the Gospel of Luke in addition to the Book of Acts.
For example, in the Book of Acts, Luke is quoting Paul:
Acts 20:35 (NASB)
35 “In everything I showed you that by working hard in this manner you must help the weak and remember the words of the Lord Jesus, that He Himself said, ‘It is more blessed to give than to receive.’”
None of the Gospel accounts (to include the Gospel of Luke) ever mention this saying by Jesus, except by Paul (recorded by Luke in the Book of Acts). So Paul learned of this saying from some source other than the information we have today contained in the Gospel accounts. That is to say, the Gospel accounts were not exhaustive summaries regarding the life and ministry of Jesus on earth, and of course (as the above verse demonstrates) the Apostle Paul had relied on some source regarding the quote from Jesus (other than the information we have today contained in the four Gospel accounts).
In this regard, the Apostle John closes his Gospel account with the following caveat, which is to suggest that no account could capture an exhaustive compilation of the life and sayings of Jesus, because they were so prolific.
John 21:25 (NASB)
25 And there are also many other things which Jesus did, which if they were written in detail, I suppose that even the world itself would not contain the books that would be written.
In other words, Paul had direct access to the life and sayings of Jesus through his direct personal contact with Peter (Gal 1:18) and James (Gal 1:19) in addition to others such as John Mark and Barnabas, who introduced Paul after his conversion to "the apostles" in Jerusalem (Acts 9:27). Thus Paul had extensive personal (verbal) contact with people who had direct contact with Jesus and His sayings.
Notwithstanding that Paul also had extensive personal contact with Luke (during his second missionary journey) and with the author of the second Gospel, John Mark (during his first missionary journey), there is parallel in the writings of Paul to indicate that he had familiarity with exclusive information found in the Gospel of Matthew, which does not appear in any of the other Gospel accounts.
Matthew 11:25-26 (NASB)
25 At that time Jesus said, “I praise You, Father, Lord of heaven and earth, that You have hidden these things from the wise and intelligent and have revealed them to infants. 26 Yes, Father, for this way was well-pleasing in Your sight.
Paul makes the same contrast in his first epistle to the Corinthians.
1 Cor 1:26-29 (NASB)
26 For consider your calling, brethren, that there were not many wise according to the flesh, not many mighty, not many noble; 27 but God has chosen the foolish things of the world to shame the wise, and God has chosen the weak things of the world to shame the things which are strong, 28 and the base things of the world and the despised God has chosen, the things that are not, so that He may nullify the things that are, 29 so that no man may boast before God.
These verses provide the most striking parallel between Paul and Matthew, which is not found in the other Gospel accounts. That is, the Lord hides wisdom from those who are “always learning and never able to come to the knowledge of the truth” (cf. 2 Tim 3:7), and instead discloses wisdom to those who are humble and who love Him.
Paul had direct contact with "the apostles," and so he had personal (verbal) access to people who had walked and talked with Jesus. However, there is one exclusive example where information in the Gospel of Matthew appears in the writings of Paul, which is not found in any other Gospel account. This parallel suggests that Paul had direct personal contact with Matthew and/or to the written information that today forms the Gospel of Matthew. In conclusion, Paul's sources of information concerning the life and sayings of Jesus appear to have been various, which would include Matthew and/or his writings. This wide-swathe access to various first-hand accounts would have been beyond the scope of what we read in the Gospel accounts, as is evident by Paul's quote of Jesus in Acts 20:35, which is not found in the four Gospels.