Allardyce stabilised the club and helped them finish the campaign in a respectable eighth position.
Although Allardyce successfully lifted the Toffees away from the relegation zone and up to eighth in the standings, his defensive style was criticised by fans, who booed the team regularly towards the end of the season.
Director Professor Denise Barrett-Baxendale said: "On behalf of the chairman, board of directors and [owner] Mr Farhad Moshiri, I'd like to thank Sam for the job he has done at Everton over the last seven months".More news: Pirates' Marte leaves due to side injury
Everton have also been interested in Shakhtar Donetsk coach Paulo Fonseca, although he is also a contender to replace David Moyes at West Ham.
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. 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.
Wayne Rooney's future at the club was reported to have rested on whether Allardyce would be manager next season.
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.More news: New 'Mission: Impossible - Fallout' trailer is the most death-defying yet
Allardyce, who passed 1,000 games in charge of an English club in January, said he was "disgusted" by the club's decision to make the changes public without his knowledge, according to the Liverpool Echo.
He was said to be "disgusted" at the club, and said: "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".
Asked about Everton's current situation, Stubbs told talkSPORT host Jim White: "I'm pretty sure they have someone in mind".More news: 'SHE'S PARANOID': Hillary Clinton WARNS Australia of China and Russian 'INTERFERENCE'