West Ham boss David Moyes will not be keen to return to Everton in a hurry.
West Ham remain 18th, are winless in seven and have taken one point from three under Moyes, who has now lost five out of five meetings with Everton since leaving them in 2013.
David Unsworth has encouraged the public to give the next manager of Everton time and patience to build the club.
With Unsworth overseeing his final match at home to West Ham tonight, Allardyce's first game in charge will be against Huddersfield Town at Goodison Park this Saturday.
Former England boss Sam Allardyce is being strongly linked with the vacancy and talks are said to be taking place between the club and his representatives.
Allardyce has been out of work since resigning as Crystal Palace managerin May. "I have no ambitions to take another job".
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"It's a great night, not for me but for the club", Unsworth said.
Though that goal failed to open the game, Everton doubled their lead ten minutes later.
That victory, when the Toffees came from two down late on to secure a 3-2 victory over Watford, looked like it might be a turning point.
Everton, now 17th in the table, have won just one of the seven games under Unsworth, losing five, and fan opinion turned sharply with their last two outings - a 5-1 home loss to Atalanta and a 4-1 reverse at Southampton.
The club's prospects had appeared so different at the beginning of the season with Everton having invested heavily following last season's impressive seventh-place finish.
Everton's chief problem is clear to see.
The Toffees spent more than £130m on new players in the summer but have struggled without last season's top scorer Romelu Lukaku, who was sold to Manchester United for £75m in July.