But Moyes, 49, will clearly find it hard to quit Everton, where he enjoys such a strong relationship with the chairman Bill Kenwright.
The former Preston boss said: "Everton now has a stability which many clubs would, I am sure, welcome.
"We have a chairman and a board who run the club the best they possibly can.
"They don't constantly change the manager or have a big turnaround of players, because undoubtedly Everton would find themselves in financial trouble if that was the case.
"I believe the stability we have had is one of the reasons that we have been consistent challengers at the right end of the Premier League table for some time now."