I'm not sure whether anyone has identified much in the way of oral traditions specifically regarding the sayings of Jesus that can be found in non-gospel literature, but I think it's probably a fruitful study.
While detouring a bit from Acts on Sunday, we read some in 1 Corinthians 13, the famous love chapter. In it, Paul writes
2 [I]f I have all faith, so as to remove mountains, but have not love, I am nothing.1 Corinthians is usually dated ca. AD 54, well before the gospel of Matthew (generally dated ca. 80–100), in which this famous saying of Jesus is recorded:
For truly, I say to you, if you have faith like a grain of mustard seed, you will say to this mountain, ‘Move from here to there,’ and it will move, and nothing will be impossible for you.(Mt 17.20, cp Mt 21.21)
I would probably reject the hypothesis that Matthew's story somehow relies on the circulation of the letter of Paul to the Corinthians: the provenance of the letter and its distribution seem unlikely to me to have influenced the writing of the Matthew.
It seems rather more likely that the gospel narrative and Paul's almost off-hand reference are part of a broader oral tradition that undergirded the nascent Christian church.
Acts records another such instance, this time without a corresponding gospel passage, when Luke quotes Paul in his final address to the Ephesian church (probably somewhere around AD 60, maybe AD 61):
In all things I have shown you that by working hard in this way we must help the weak and remember the words of the Lord Jesus, how he himself said, ‘It is more blessed to give than to receive.’(Acts 20.35)
I think it would be an interesting study to look at these pregospel letters to try to piece together some other examples of oral testimony, and perhaps I'll take that on as a project this year.
