Moyes has insisted that his star man Marouane Fellaini is not set to leave Everton at the end of the season, and hopes to bring in Strootman to play alongside the big Belgian.
But it appears there is some competition for the versatile Dutch international, who can play as a holding midfielder, deep-lying playmaker or box-to-box central midfielder.
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Manchester United have been previously linked with the 23-year-old, and reports on the continent indicate that Sir Alex Ferguson hasn't ruled out a move for him this summer.
AC Milan are also interested in Strootman, who is open to the idea of leaving the Dutch Eredivisie ahead of next season.
United want the former Utrecht man to replace veteran pass-master Paul Scholes, while Milan could include midfielder Urby Emanuelson - currently on-loan at Fulham - in any deal.
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