Marcus Rashford has suffered a double stress fracture to his back, ruling him out for up to three months.
The Athletic says United had already enquired about the availability of Piatek prior to Rashford's injury, and were hopeful of securing the Polish star on an initial loan deal.
Piatek was a prolific marksman for former club Genoa, but has struggled to hit the same heights for Milan since.
With only five goals to his name in 19 appearances this season, Milan are now said to be prepared to let Piatek leave the club.
The Rossoneri are driving a hard bargain however, demanding a fee in the region of €35million (£29.7m).