If you don’t know, there is a thing called the “agrapha,” which is sayings attributed to Jesus not found in the canonical Gospels.
This includes things like Acts 20:35: "Remember the word of the Lord Jesus, how he said: It is a more blessed thing to give, rather than to receive."
There are lots of sayings attributed to Jesus in the “apocryphal” Gospels, among the church fathers, and in various papyri fragments such as P. Oxy 840 and P. Eg. 2.
Recently, a new fragment has been published which is P.Oxy 5575, which derives from finds in Oxyrhynchus in Egypt. The text says:
. . . he died [I tell] you: [do not] worry [about your life,] what you will eat, [or] about your body, what [you will wear.] For I tell you: [unless] you fast [from the world,] you will never find [the Kingdom,] and unless you . . . the world, you [will never . . .] the Father . . . the fowls, how . . . and [your] heavenly Father [feeds them] You [also] therefore . . . [Consider the lilies,] how they grow . . . Solomon . . . in [his] glory . . . [if] the Father [clothes] grass which dries up and is thrown into the oven, [he will clothe] you . . . You [also] therefore . . . for [your] Father [knows] . . . you, [Instead] seek [his kingdom, and all these things] will be given to you] as well. [Translation: J. Fish, D.B. Wallace, and M. W. Holmes].
Now what is interesting here, is that this text parallels Matt 6:25-26, 28-30, 32-33, Luke 12:20, 22, 24, 27-28, 30-31, and Gospel of Thomas 63, 36, 27.
P. Oxy 5575 is, I think, clearly dependent upon the Synoptic tradition (probably Matthew in particular) but it retains a degree of flexibility that intersects with and perhaps is interweaved with other versions of the Jesus tradition as found in the Gospels of Luke and Thomas.
You can read more about it on the sites below:
Candida Moss, "Scholars Publish New Papyrus with Early Sayings of Jesus" (Daily Beast)
Peter Gurry, Synopsis of P.Oxy. 5575, Matt, Luke, and Thomas (Evangelical Textual Criticism Blog)
Mark Goodacre, English Synopsis of P.Oxy. 5575, Matthew, Luke, and Thomas (PDF)
Brent Nongbri, "The Date of the New Oxyrhynchus Sayings of Jesus P.Oxy. 87.5575", Variant Readings Blog.
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"P. Oxy 5575 is, I think, clearly dependent upon the Synoptic tradition (probably Matthew in particular) but it retains a degree of flexibility that intersects with and perhaps is interweaved with other versions of the Jesus tradition as found in the Gospels of Luke and Thomas."
So you do not see as a possibility, albeit perhaps remote, that the tradition found in the other Gospels may have been dependent on this source's tradition?