Allardyce was appointed as successor to Ronald Koeman on and 18-month deal last November, with an immediate brief of guiding Everton away from relegation danger.
Marco Silva is thought to be the frontrunner for the Everton post once more, despite their approach for the then-Watford boss a year ago causing acrimony between the two clubs and preceding the Portuguese coach's departure from Vicarage Road. Silva's Portuguese compatriot Paulo Fonseca, the manager of Shakhtar Donetsk, is second favourite.
However, he was booed by fans following his side's 2-1 defeat at Burnley in March, and was again criticised after their draw with Swansea six weeks later. The 63-year-old also did not see eye to eye with the board, who sent out a survey to the fans asking them to rate his performance without getting his approval to do so first. The club made the announcement while the former England boss was at the League Manager's Association 26th Annual Awards in London.
5 people dead as severe storms lash Northeast
Northeast on Tuesday, snarling transport and causing widespread power outages in an area home to tens of millions of people. A man in CT was killed when a tree fell on his truck, Danbury Mayor Mark Boughton told the Hartford Courant .
England's all-time top scorer had reportedly agreed terms on a £12.5 million move to DC United in Major League Soccer last week.
Professor Denise Barrett-Baxendale was promoted to chief executive and Allardyce said he was "disgusted" not to have been informed.
"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", said Allardyce.
As part of the questionnaire, supporters were asked whether they agree with the statement: "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".