Sam Allardyce has left his role as Everton manager after six months.
Everton, who were 13th in the Premier League when Allardyce arrived, finished the season in eighth.
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.
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”.
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.
However, Allardyce was booed by fans following his side’s 2-1 defeat at Burnley in March, and he was again criticised after their draw with Swansea six weeks later.
Everton under Allardyce
|Everton under Allardyce in the Premier League||Stats||Rank|