The 63-year-old signed a contract until June 2019 when he took over in November following the sacking of Ronald Koeman.
Everton, who were 13th in the Premier League when Allardyce arrived, finished the season in eighth.
Allardyce has been heavily criticised by fans for his management and Everton’s style of play since he took over and the club said the decision was part of their “longer-term plan”.
Everton chief executive Denise Barrett-Baxendale thanked Allardyce, adding: “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.
“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.”
It is understood former Hull City and Watford boss Marco Silva is the main candidate to succeed Allardyce.
The 40-year-old Portuguese was owner Farhad Moshiri’s first choice when he dismissed Dutchman Koeman.
Everton have also been interested in Shakhtar Donetsk coach Paulo Fonseca, although he is also a contender to replace David Moyes at West Ham.