Caretaker boss David Unsworth has stressed Everton's next permanent manager will need "time, patience, support and backing".
It seems the Toffees will soon have a new man in charge, with Unsworth on Tuesday saying "talks are ongoing" between the club and "a few people".
Negotiations are continuing between Everton and Sam Allardyce's representatives.
Sky Sports News revealed that Everton had also held talks on Friday with the highly-rated Shakhtar Donetsk head coach Paulo Fonseca.
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Unsworth, who stepped up from the U23s following Ronald Koeman's sacking on October 23, said he would still be in charge for Wednesday's clash with West Ham as far as he was aware.
Asked for his understanding of the managerial situation, he said: "As we sit here right now, I believe the board are talking to a few people.
"Talks are ongoing, but nothing has been finalised yet with anybody. So I continue to do what I was asked to do five weeks ago."
Asked directly about Allardyce, Unsworth said: "Sam is a top manager, but I think it would be wrong of me to talk about individuals.I have seen the names that have been bandied about - all top managers. Whether there's truth in the names of some, or all, or none, I don't know.
"But whoever gets the honour of being Everton manager will have my full support and my full backing."
Everton have won only one of the seven games of Unsworth's temporary spell to date and lost five of them, with their last two outings a 5-1 home loss to Atalanta and 4-1 reverse at Southampton.
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Unsworth added: "I think we need our best players fit, quickly, some proven Premier League recruitment in early January and I think we need to win a couple of games and get confidence flying again.
"When confidence is low, it takes a bit of time to build that back up. I don't think there's a quick fix - I think whoever gets the job needs time, patience, support and backing."
The Toffees are now just two points clear of the bottom three after they were beaten 4-1 at Southampton on Sunday.