Sources told Football Insider.
The club’s directors are grateful to Sean Dyche for keeping the team alive last season, while some believe Moyes is capable of taking his former club to the next level.
The fans are also said to be key to any managerial changes at Goodison Park in the near future.
Sources have told Football Insider that Dyche could be sacked before the next international break if results continue to be poor and fans then turn against the manager in large numbers.
The Merseyside giants are currently bottom of the Premier League table after failing to win any of their opening three matches.
Everton have conceded 10 goals in those games and the club squandered a two-goal lead in the 86th minute in a 3-2 loss to Bournemouth last weekend.
David Moyes could make a sensational return to Everton
Football Insider previously revealed that Dyche is under increasing pressure at Everton, while Moyes would be “happy” to return to his former club.
It wouldn’t be the first time the Scottish manager has returned to the club, having spent two separate spells in charge of West Ham.
Moyes managed Everton from 2002 until 2013, when he left the club to take over from Sir Alex Ferguson at Man United.
Moyes guided Everton to the Champions League in the 2004-05 campaign, establishing the team as a club that would constantly challenge for a top seven finish in the Premier League.