The Scot, appointed in November as successor to Slaven Bilic, steered West Ham to Premier League safety in a campaign capped by two goals from Manuel Lanzini and Marko Arnautovic's 11th of the league season rendered Oumar Niasse's strike a consolation.
The leading candidates to follow Moyes in the West Ham hotseat are as follows.
Moyes fulfilled his brief of keeping the Irons in the Premier League this season.
He took cost of Shakhtar Donetsk two years in the past in what was his first overseas job and gained the title in Ukraine in his opening two seasons, finishing a league and cup double in that first marketing campaign.
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Allardyce admitted after Sunday's final day of the season defeat to former club West Ham he could not be confident over his future.
West Ham came out of this battle with the points but both sets of fans know they've got a lot of work to do in the summer to stand any chance of getting to where they think they should be - in Europe. However Arsenal are likely to have first dibs on the Spaniard, who spent much of his playing career in the Spanish second division.
The feeling at West Ham amongst the owners David Sullivan and David Gold is that they need a manager who can take the club further forward, to the next level.
A number of other names have been linked with the vacancy in recent days, including Ronald Koeman, Carlo Ancelotti and Marcelo Bielsa.