Everton sit bottom of the table under manager Sean Dyche with three straight defeats from the opening three games.
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Moyes is being discussed as a replacement if Everton chooses to fire Dyche.
Moyes, who left West Ham after last season, is said to be interested in returning to Everton, says Football Insider.
The 61 year-old was manager at Everton between 2002 and 2013. He has since been at Manchester United, Real Sociedad, Sunderland and West Ham.