Chairman Bill Kenwright and a significant number of Everton's players are understood to want the club's U23s manager David Unsworth to get the job full-time, Sky Sports News understands.
Unsworth has today been confirmed in temporary charge for the second time, having held the reins in 2016 after the departure of Roberto Martinez.
The 44-year old is highly regarded at Goodison Park and will be in the dugout starting with Wednesday's Carabao Cup match at Chelsea.
Chairman Kenwright is well aware that Everton's finest hours post-war have come when Everton have been managed by a former player, including the likes of Joe Royle, Howard Kendall and Harry Catterick.
Owner Farhad Moshiri, however, is thought to prefer a more established manager, who has experience at the highest levels of European football, with former Bayern Munich boss Carlo Ancelotti fitting that bill.
Sky Sports News understands Burnley's Sean Dyche and the former Man City manager Manuel Pellegrini, who is currently managing Hebei China Fortune, are the other front-runners.
It is understood that Everton were yet to make contact with the Burnley manager's representatives, but Leicester appeared very keen to lure Dyche to the King Power Stadium.