Moyes is succeeding Sir Alex Ferguson at Manchester United this summer and that has left Everton chairman Bill Kenwright looking for a new boss for the first time in 11 years.
It is understood that the club are considering several options, with Phil Neville, David Weir and Alan Stubbs three of those in contention.
Neville's Everton playing career is coming to a close after over 300 appearances in eight years, and the 36-year-old has been interviewed for the vacancy.
Former defensive duo David Weir and Alan Stubbs, who are currently on the coaching staff on Merseyside, have also been spoken to about the job.
There are five external candidates, with Porto coach Vitor Pereira the current favourite for the position with Sky Bet.
Pereira was formerly Andre Villas-Boas' assistant at Porto before taking charge in 2011 and he has won the Portuguese title in both seasons at the helm.
He is out of contract at Porto this summer and has emerged as a leading candidate to replace Moyes, along with Wigan Athletic boss Roberto Martinez.
The Spaniard spoke to Wigan chairman Dave Whelan earlier on Wednesday and he is expected to make a decision on his future in the next 24 hours.
Martinez led Wigan to FA Cup final success earlier this month before seeing the club relegated from the Premier League and he has also been linked with the Stoke City vacancy.
Swansea City manager Michael Laudrup, Cardiff City boss Malky Mackay and former Chelsea head coach Roberto Di Matteo are also in the frame.
Laudrup recently confirmed he has no plans to leave Swansea, while Mackay has also played down speculation he could take over from Moyes.