Marco Silva and Paulo Fonseca are among the early candidates in line to replace Allardyce.
"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".
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.
The 40-year-old Portuguese was owner Farhad Moshiri's first choice when he dismissed Dutchman Koeman.
Allardyce, whose previous clubs include Bolton, Newcastle, Sunderland and Crystal Palace, landed his dream job with England in 2016 but that turned sour after only one match, with Allardyce stepping down following a newspaper sting.
Mission: Impossible - Fallout Trailer Proofs That Tom Cruise Still Got It
A tiny clip to tease the release of the trailer was poster on Twitter by the franchise's official page. But still never Tom cruise has not appeared in any superhero movie any Kynoselen (or outside).
Court fight over Iowa's new 'heartbeat' abortion law begins
Kim Reynolds and was immediately met with threats of a lawsuit from abortion groups who claim the ban is clearly unconstitutional. The law contains certain ultrasound requirements, though Tuesday's lawsuit only challenges the heartbeat ban portion of the law.
Legislature interviews Underwood, others as potential successor to AG Schneiderman
Kathleen Rice (D-LI), who announced Tuesday she would be running for re-election instead. But she won't have to contend with Rep.
The Premier League subsequently instigated a mediation meeting for the two clubs last month but the issue failed to be resolved, and with the dispute having dragged on for four months the league could be forced to find a conclusion to the issue themselves.
Wayne Rooney next club odds: Where could Everton star go?
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.
A statement was released on Everton's website this morning to announce the decision with chief executive Denis Barrett-Baxendale stating the decision was taken "as part of our longer-term plan".
'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, ' he had said.