Moyes will become Manchester United's new manager on July 1 on a six-year contract to replace the retiring Sir Alex Ferguson.
Here, we take a look at who could be in contention to take over at Goodison Park.
ROBERTO MARTINEZ (Sky Bet odds: 7/2)
The Spaniard, who interested Liverpool last summer, has won many admirers with his brand of football at Wigan and his ability to work on a budget may appeal to the Goodison hierarchy.
NEIL LENNON (8/13)
The Celtic boss may feel he has done all he can in Scotland, especially with the loss of Old Firm games. Has European experience, having beaten Barcelona this season, which would appeal to the Toffees.
VITOR PEREIRA (12/1)
The 44-year-old has enjoyed two hugely successful seasons since succeeding Andre Villas-Boas as Porto coach and is out of contract in the summer, which means Everton would not have to pay compensation. Unknown quantity in British football, however.
MICHAEL LAUDRUP (20/1)
Winning a trophy in his first season in English football has only enhanced the Dane's reputation, having been lauded for his work with the Swansea side in the short term. Has managerial experience in four countries in addition to a glittering playing career.
PHIL NEVILLE (9/1)
The Toffees captain is scheduled to leave the club to extend his playing career but could possibly be tempted to stay if it was as manager, having turned down a job on the coaching staff. Will assist England Under-21s this summer but would be a leap of faith by the Everton board.
MARK HUGHES (33/1)
Hughes, a former Everton player, has already expressed his interest. Sacked by QPR in November, his appointment would likely be unpopular among the majority of fans.