Keisuke Honda is reportedly set to leave Milan this summer
Toffees manager Roberto Martinez is said to be keen on strengthening his squad before the transfer deadline shuts.
And Honda has emerged as a possible target after reports in Italy claimed Milan are open to selling the midfielder.
Japan international Honda, 29, is also wanted by Tottenham, Italian newspaper Tuttosport suggest.
The dynamic player only arrived at the San Siro two years ago but has been given no assurances of his place in the side for the coming season.
It is claimed his arrival at Milan was in fact no more than a marketing scheme to attract supporters from the Far East.
Tuttosport outline that the player has not taken kindly to new manager Sinisa Mihajlovic pre-season training regime and is seeking a move way from Italy.
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Spurs, who are yet to add to their midfield this summer, are understood to be eyeing up a bid to rival Everton's if the player did become available.
Honda has represented Japan at the previous two World Cup finals and helped the nation to a victorious Asian Cup title in 2011.