Hodgson set to remain England boss until 2016
Questions over the head coach's future arose after England's tournament status was left hanging by a thread following Thursday's 2-1 loss to Uruguay.
With Hodgson's men having been defeated by the same scoreline against Italy on Saturday, England now need every remaining Group D result to go their way to have any hope of progressing.
Yet despite those disappointing results, Dyke has revealed Hodgson - who said immediately after the Uruguay game he had no intention of resigning - will remain at the helm until at least 2016.
He told Sky Sports News: "We're supportive of Roy Hodgson, we've asked him to stay as manager.
"We were not humiliated. They were narrow defeats. We don't see any great value in changing Roy. He's done a good job."
Hodgson has been held the role since just before UEFA Euro 2012, a tournament in which England were eliminated in the second round by Italy.