Ex-Everton boss Moyes could return to Goodison Park in the interim role; Arsenal also interested, say Sky in Italy.
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.
Arsenal are also considering Ancelotti, who managed Napoli for 19 months, to become their next manager, according to Sky in Italy.
Napoli announced the decision to dismiss Ancelotti hours 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 the 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.”
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.