Moyes, who returned to the Toffees last season, has a current deal to 2027.
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The appointment was initially viewed as a short-term prospect, however such has been the Scot's impact that the board are now moving to extend his deal.
The Guardian says new contract talks are set to kickoff with the aim of extending and improving Moyes' current arrangement.
Everton sit eighth on the Premier League table as they make a push for European qualification.