This question is addressed by John Nelson, who begins by noting several factors that might suggest some distance between what Jesus actually said and what is recorded in the Gospels:

First, Jesus spoke Aramaic, but the Gospels were written in Greek. This means that we often depend (at best) upon a translation of Jesus’ original teaching.

Second, the Gospels were written in an oral culture. This means that many of Jesus’ teachings were likely passed down verbally before they were written down.

Third, the Gospels are ancient biographies. And ancient biographers and historians often granted themselves literary license with their subject’s words.

Finally, we find various discrepant teachings in the Gospels. These discrepancies inhere in both the content and the style of Jesus’ teaching.

(Obviously, the use of the word discrepancies rather begs the question, but I will let it rest for the moment).

What reasons, then, do we have for supposing that the (Synoptic) Gospels record the actual words of Jesus (substantially, if not completely)?

(a) Recurrent motifs.  A number of common themes occur across the Gospels in different settings and contexts.  For example: teaching about the kingdom of God, about religious hypocrisy, about Jesus as the Son of Man, and about his impending death and resurrection.  Much of this teaching is in the form of brief sayings, anecdotes and stories.  And while these do not necessarily  always capture the precise words of Jesus, it is likely that they carry the gist.

(b) Oral tradition.  Bart Ehrman has likened the process of oral tradition to the Telephone Game (aka Chinese Whispers).  The whole point of the game is to demonstrate that a line of oral tradition becomes increasingly corrupt with repeated retellings.  But the analogy is flawed, because it fails to take into account the stability of the oral traditions behind the Gospels (which reflect what Dunn calls ‘variation within the same’).  Nor does it take into account the memorability of many of the sayings attributed to Jesus (Robert McIver refers to the telling of a joke – it may be told using different words, but the gist will be retained).

(c) Distinctive genius.  The body of teaching ascribed to Jesus in the Synoptic Gospel is of such a nature that it is best considered as the thought of a single unique mind.

(d) Palestinian context.  The teachings of Jesus show detailed knowledge of Palestinian geography and customs:

‘For example, Jesus’ teaching knows corban, the authentic word for a Temple tax, that one would ‘go down’ to Jericho (well below sea-level; Lk. 10:30), and assumes the Temple in Jerusalem was still standing (e.g. Lk. 5:14).’

Jesus’ teaching is also notable for what it does not include.  For example, it does not reflect the controversy about baptism, which became prominent later.

Conclusion

Nelson’s own conclusion is that the words of Jesus have not been entirely lost.  That, it seems to me, is something of an understatement, given his own argumentation.