The Scot’s future at Goodison Park is far from certain, with negotiations regarding a new deal at the club having stalled.
Moyes’ stock is at a real high, leading to speculation that he could take charge of Chelsea ahead of next season, or move abroad to manage a top European club.
Everton are aware that their man could choose to leave, and have drawn up a list of potential replacements.
Martinez is said to be top of the pile, with the Toffees’ hierarchy impressed by his work at Wigan.
The Spaniard has kept the Latics in the Premier League against all the odds for some time now, playing attractive football along the way.
Despite having rejected approaches from both Aston Villa and Liverpool during the past two off-seasons, Everton are confident that they can tempt their target to move on in search of a new challenge.
Martinez is widely regarded as the man who built the current Swansea side, having taken control of the club during their time at the lower end of the domestic footballing ladder, guiding them up the divisions.
Everton are thought to be keen on the style of football he brings, and are optimistic that he can get the very best out of their talented squad.
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