The Gospels were written in a predominantly hellenistic environment, and they were written in Greek. But Greek was not the native language of their central Figure, nor of the earliest apostles, if it was not unfamiliar to them. Jesus must have conversed in the Galilean dialect of Aramaic, and His teaching was probably almost entirely in Aramaic. At the basis of the Greek Gospels, therefore, there must lie a Palestinian Aramaic tradition, at any rate of the sayings and teachings of Jesus, and this tradition must at one time have been translated from Aramaic into Greek. Some have thought that the Evangelists themselves were the translators of these Aramaic sources of the Gospels; they certainly must have utilized, if they did not themselves translate, early translation sources. The 'Aramaic problem' of the Gospels is to determine, by internal evidence, to what extent the Greek Gospels are written in or embody 'translation Greek' or how much Aramaic influence can be detected in them.
from page 16, Aramaic Approach to the Gospels and Acts