The Mirror says United had scouts posted to check on the teen in last week's Champions League tie with Real Madrid.
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Real Madrid came away with a 5-4 win, though Mouzakitis impressed on the night.
Indeed, the youngster's performance drew praise from Real coach Xabi Alonso and sparked talk of a potential offer being prepared by the Spanish giants.
United are also keen, with management pair Jason Wilcox and Christopher Vivell working on rejuvenating the squad's midfield options.