The Toffees sacked Ronald Koeman after the 5-2 thumping by Arsenal at Goodison Park left them in the bottom three, with just eight points from nine games.
Dyche is wanted by Leicester, after the Foxes sacked Craig Shakespeare this month, but Everton are hoping they can tempt the Burnley manager to Merseyside instead.
Indeed, Dyche said last week that he was “open-minded” about his future.
Under-23s’ manager David Unsworth, alongside assistant John Ebbrell, will take temporary charge of Everton for their Carabao Cup clash with Chelsea tomorrow night – and, despite their interest in Dyche, the club’s board have not ruled out giving the former defender a chance to prove himself.
Unsworth’s stock is high at Everton after he guided the U23s to the Premier League 2 title last season and he brought through Tom Davies, Dominic Calvert-Lewin, Jonjoe Kenny and Mason Holgate to the brink of the first team.
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He has been in charge for one game before, a 3-0 win over Norwich City on the last day of the 2015-16 season following the sacking of Roberto Martinez.
The former defender is seen as having management potential but the vacancy may have come too early for him this time.
Last night, former Manchester United star Ryan Giggs put his name forward but Carlo Ancelotti is also available, having been sacked by Bayern Munich last month, and he is admired by Everton’s major investor Farhad Moshiri.
Moshiri and Everton chairman Bill Kenwright decided to dismiss Koeman after a nightmare start to the season that has seen them lose seven of their last 11 games.
Koeman did not seem worried about his position when he dined at Manchester’s San Carlo restaurant on Sunday with his wife, Bartina, with Manchester City boss Pep Guardiola also at the venue.
He arrived at the club’s Finch Farm base yesterday morning and took training as normal, having confided in friends he would be given time to turn things around at the club.
But he was later informed that would not be the case and his £6million-a-year contract was terminated with immediate effect.
Koeman’s assistant coach and brother Erwin, goalkeeping coach Patrick Lodewijks and fitness coach Jan Kluitenberg will also depart and were last night negotiating their severance packages.
