Dyke said that the FA believe that there was no racist intent from Hodgson and that he is an "honourable man" in the wake of a leak from the England camp that claimed the manager inappropriately referenced a monkey joke in reference to Andros Townsend.
"Roy Hodgson is a man of the highest integrity, an honourable man who is doing a great job with the England team. He has and deserves the full support of the Football Association," the statement said.
"He has fully explained to us what he said and the point he was making to the players in the dressing room at half-time on Tuesday night. He has also explained the context in which he made his remarks.
"He has made clear there was no intent to say anything inappropriate, and he was certainly not making any comments with any racist connotation. Importantly, he has apologised for any unintended offence that may have been taken."
Hodgson has been in charge of the England team since 2012.