The Blues’ boss meets owner Farhad Moshiri next week for talks about his position, with fans clamouring for his departure.
Everton have already made enquiries about the availability of ex-Hull and Watford boss Marco Silva and Shakhtar Donetsk’s Paulo Fonseca.
Allardyce’s position will have direct implications for Wayne Rooney’s future at the club.
On a day of farce, the former England skipper was left out for Everton’s final game of the season at West Ham on Sunday because of a knee injury… soon after being declared fully fit by Allardyce.
With Rooney given permission to speak to Major League Soccer’s DC United by Moshiri, Allardyce confirmed on Friday morning that he would speak to the player at the training ground “to see where the situation lies”.
The manager also insisted: “He’s fit and in training, yes. You’ll find out if he is playing on Sunday,” he said.
But just an hour later, Rooney was heading out of the Finch Farm complex following his scheduled meeting, just as the other players began training.