Everton will begin contract talks with Carlo Ancelotti after he was sacked as Napoli manager on Tuesday evening.
Former Everton boss David Moyes could still return to Goodison Park in an interim role, while a new permanent manager is put in place, Sky Sports News understands.
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Arsenal are also considering Ancelotti, who managed Napoli for 19 months, to succeed Unai Emery, according to Sky in Italy.
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Napoli announced the decision to dismiss Ancelotti an hour after he led the club to the knockout stages of the Champions League with a 4-0 home win against Genk, ending a nine-match run in all competitions without victory.
A Napoli club statement read: "SSC Napoli has decided to revoke the role of first-team coach from Mr Carlo Ancelotti.
"The rapport of friendship, admiration and reciprocal respect between the club, its president (Aurelio De Laurentiis) and Carlo Ancelotti remain intact."
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Vitor Pereira, who was viewed as the leading candidate at Everton, pulled out of the process to become the successor to Marco Silva earlier on Tuesday.
Duncan Ferguson is currently Everton's interim manager and won an emotional first game in charge, beating Chelsea 3-1 at Goodison Park.
Sky in Italy reported on Tuesday former AC Milan manager Gennaro Gattuso is the leading candidate to take over at Napoli, and is set to sign a provisional six-month deal.
They are also understood to have enquired about Lazio boss Simone Inzaghi, who has led the club to third place in Serie A.
However, he is believed to be happy to stay at Lazio - although he would be interested in moving to England in the long-term.
Lampard: 'Classy' Ancelotti can manage anywhere
Chelsea head coach Frank Lampard says his former boss Ancelotti is good enough to manage anywhere.
Speaking after the 2-1 win over Lille, Lampard was asked if the Italian could return to the Premier League.
He said: "I've got huge respect for him. I'm sorry to hear that [he has been sacked].
"I know it has been a tough situation for him lately. I saw his interview before the game and it was typically classy in what seemed like a tough time for him. He's always handled himself brilliantly.
"I think he can manage anywhere in the world at the top level, of course."
Beckford: Ancelotti would be fantastic
Jermaine Beckford thinks Ancelotti would be a "fantastic fit" at Everton, but is unsure he would want to go to Goodison Park.
Beckford, who played 40 times for the Toffees, believes that although he'd be good fit at the club, there are probably more appealing jobs available to the Italian at the moment.
"I think he'd be a fantastic fit. Do I think he'll go? I don't think he will," he said.
"Not when you look at the jobs available at the moment. I would love to see him there. He's tried and tested and he knows what the league's all about."
Beckford, who scored against Ancelotti's Chelsea side in 2010, believes the way he sets his teams up is what both Everton and Arsenal need.
"[His teams] are tough to break down," Beckford said. "They tend to play with a lot of confidence as well. That's something that both Everton and Arsenal are lacking."
"But he's got to be there for a project, long-term. You can't jump into a relegation battle with his CV, that not necessarily what he's good at."
Analysis from Sky Sports' Ben Grounds
'Replicating the trophy-laden eight-year stint as AC Milan manager has always been his goal.
'After his troubled ending at Napoli, Ancelotti may opt to take some time out of the game to evaluate his next chapter, and never has the Premier League managerial landscape at the top felt more volatile.
'While the infrastructure and the draw of managing again in London would point towards Arsenal, he will be cautious of how difficult Unai Emery found replacing Wenger.
'Were Ancelotti to make a sensational move to Merseyside, he would be returning to a club with a "family" culture as experienced at the San Siro.'
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