A statement on West Ham's website confirming Moyes's departure said the club wanted to take a "different direction". "Having taken stock of the situation and reflected now the campaign is complete, we feel that it is right to move in a different direction".
The 63-year-old's style of play, perceived as overly negative, was the prime source of dissent, while Allardyce's substitutions saw him abused by his own fans from the stands at Burnley during a 2-1 defeat in March.
Allardyce was told the news this morning but hit out last night after the club announced sweeping changes to its management structure.
Former Watford boss Marco Silva is the bookies' favourite to become the next Everton manager after Sam Allardyce was sacked today.
"However, we have made the decision that, as part of our longer-term plan, we will be appointing a new manager this summer and will be commencing this process immediately".
Allardyce lasted seven months at Everton after signing an 18-month deal with the northwest club in late November when the team was near the relegation zone.
Watford remain angered by Everton's repeated attempts to lure Silva to Goodison Park over a two-month period which unsettled the Portuguese boss and led to a dramatic dip in form on field - after one win was notched in 11 matches.
Fonseca, the Shakhtar Donetsk coach, has also been linked with Everton as well as West Ham, and Louis van Gaal, Andre Villas-Boas, Sergio Conceicao, Sean Dyche and Eddie Howe are other names reportedly in the frame.
The new manager will be selected by Everton's new director of football Marcel Brands and new chief executive Denise Barrett-Baxendale.
The decision - Everton were actually 13th, five points clear of the bottom three by the time Allardyce took charge - has been debated ever since.