The Red Devils are apparently tracking the 17-year-old Netherlands left-back and would like to snap him up before his price tag rises so they can groom him to replace 30-year-old
The Mirror report that Sir Alex has already sent his brother and United scout Martin to watch the 17-year-old, most recently last weekend, and run the rule over the teenager.
Willems only joined PSV from Sparta Rotterdam in the summer but is already a member of their first-team squad and is representing the Netherlands at youth level.
The report suggests Sir Alex is not confident he has a long-term option within his current squad to fill the shoes of 30-year-old Evra, so wants to bring in a younger option and bed him in to life at Old Trafford before that time arrives.
While an immediate move for Willems, who signed a three-year contract at PSV when he joined the club, may not be in the offing, the United manager is said to be keen to swoop before his price rises too much.
The Red Devils are likely to watch the player again ahead of the New Year before they make any final decision on whether to follow their interest up with a formal enquiry.
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