Former Manchester United assistant manager Ryan Giggs has also declared an interested in the position, as well as the vacant Leicester job.
The Toffees sit 18th in the Premier League after winning just two of their opening nine league games, and are bottom of their Europa League pool — Koeman paid the price yesterday.
The Dutchman led the club to a seventh-placed finish last season, his only full campaign at the helm after arriving from Southampton, but has had a torrid time of late.
Two home defeats in four days, 2-1 to Lyon in the Europa League and 5-2 against Arsenal on Sunday, and the chorus of boos which greeted each result proved the end of the line for 54-year-old.
He arrived at the club’s Finch Farm training base as usual yesterday morning but within a matter of hours was out of a job.
A statement from the club read: “Chairman Bill Kenwright, the board of directors, and major shareholder Farhad Moshiri would all like to express their gratitude to Ronald for the service he has given to the club over the past 16 months and for guiding the club to seventh place in last season’s Premier League campaign.”
That brief note did not state who would take interim charge, nor confirm the likely exit of Koeman’s brother Erwin from the staff, but U23 boss David Unsworth is expected to step up for tomorrow’s Carabao Cup clash at Chelsea.
Former Everton star Unsworth is also considered a candidate for the top job.
Burnley boss Dyche is seen as an attainable option from within the Premier League, while former Toffees favourite David Moyes, eager for a top-flight return after leaving relegated Sunderland, spent 11 largely successful years on Merseyside before an ill-fated move to Manchester United.
Giggs is looking to get into management following his stint as Louis van Gaal’s number two at Old Trafford, having been interviewed for the job at Swansea last season, and fancies either the Everton or Leicester roles.
“I think you look at those two teams, Leicester being champions two seasons ago, Everton being a fantastic club with a fantastic history, for me these are clubs I would be interested in,” Giggs said.
“However, there are a lot of coaches out there who would be interested in those positions. I’ve said all along that I’d be open to clubs which share the ambitions of myself. I want to improve clubs and improve players, to enjoy working for them, and for players to enjoy the challenge.
“There are clubs in the Premier League which would be good jobs but also in the Championship and League One. It’s more about the philosophy for me than the stature of the club.”
Whoever eventually takes over at Goodison Park will inherit a squad with just two wins and eight points from nine league matches.
They will also find a squad many consider to be poorly balanced, with a surfeit of creatives in the number 10 mould and no adequate replacement for departed frontman Romelu Lukaku.
Gylfi Sigurdsson and Davy Klaassen cost almost €80m
between them and Wayne Rooney was brought back from Old Trafford on a high-profile free transfer, but Lukaku’s firepower is sorely lacking.
A further €60m was spent on goalkeeper Jordan Pickford and defender Michael Keane, yet the Toffees have conceded 18 goals in nine league games and only rock-bottom Crystal Palace have a worse goal difference.
Leon Osman, a Toffees stalwart between 2000 and 2016, believes Koeman’s fate had become inescapable.
“It’s not just results, performances of late have been really below par, really worrying and it was pretty inevitable this was going to happen,” he said.