Sky Sport Italia reports that United have made a €12 million bid for the 25-year-old, although that looks likely to be rejected as Torino want at least €20m.
Darmian joined Torino in 2011 from Palermo and has scored six times in 150 appearances for the side since the transfer.
He has also played for AC Milan for three years, after he moved up to their senior side following eight years in their youth squad, in a spell which included a loan out to Padova.