The former England boss endured a fractious relationship with the club's hierarchy in recent weeks and was fuming after a re-structure of his backroom staff.
Denise Barrett-Baxendale, deputy chief officer of the club said: "On behalf of the Chairman, Board of Directors and Moshiri, I would like to thank Sam for the job he has done at Everton over the last seven months".
Sam Allardyce has been sacked by Everton after guiding the club to an eighth-placed finish.
Allardyce was hired after a dismal start to the season under Ronald Koeman saw Everton fearing a relegation battle.
Everton were unbeaten in their first six league matches under Allardyce and claimed their only Europa League win under the ex-Crystal Palace, Sunderland, Blackburn, West Ham, Bolton and Newcastle boss.
A change in manager could also affect Wayne Rooney's future at the club.
The former England striker, 32, has agreed a deal in principle that could see him leave for Major League Soccer side DC United this summer in a £12.5m deal.
Allardyce was presented with an award at an LMA dinner on Tuesday evening for reaching 1,000 games in management.
"I'm shocked, disappointed and disgusted that the football club didn't have the decency to tell me, my Director of Football and my staff about the changes".
As part of the questionnaire, supporters were asked whether they agree with the statement of "I have a high level of trust in the current manager and coaching staff at Everton e.g.in making the right decisions to get the best out of the team".