"Sam was brought in at a challenging time last season to provide us with some stability and we are grateful to him for doing that".
Wayne Rooney's future at the club was reported to have rested on whether Allardyce would be manager next season.
'Again, we'd like to place on record our honest thanks to Sam for his work with us over the last few months and wish him well for the future'.
So who will take over?
Thompson feels the unrealistic expectations created by Everton's spending last summer ultimately cost the former Bolton boss his job at Goodison Park at the end of the season.
The club is said to have contacted both Marcos Silva and Shaktar Donetsk manager Paulo Fonseca to replace Allardyce.
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Allardyce took over the reins on November 30 previous year, six months after leaving a similarly short stint at Crystal Palace.
Should Everton press on and appoint Silva as their new manager they may be open to sanctions and could even be deducted points however it is understood that chairman Bill Kenwright is in talks with Watford and Premier League bosses so as to come to a settlement.
Watford "remain furious" that Everton's interest previous year unsettled Silva and a poor run of form led to his sacking in January.
The Goodison Park club are eager to appoint Silva as the successor to Allardyce, but are now involved in a dispute with Watford over an approach made for the Portuguese while he was still at Vicarage Road.
The Daily Express adds that former Manchester United manager Louis van Gaal has emerged as a "shock candidate" while Leicester's Claude Puel and Burnley's Sean Dyche are also in the running.
Speaking to the Mirror at the League Managers Association annual awards, Allardyce made his anger at not being informed of those decisions known and suggested the writing was on the wall regarding his own future. Moyes is also without a job after leaving West Ham today.