United have received an inquiry from an unnamed club for the 29-year-old, who arrived from Torino for £12.7m in 2015.
Darmian has played only 13 times since December 2017 and his contract is due to expire next year.
United manager Ole Gunnar Solskjaer is willing to let Darmian leave, something the Italian wanted 12 months ago.
England defender Chris Smalling and Argentina’s Marcos Rojo are other United players with question marks over their futures.
Italian side Roma are interested in Smalling, 29, but despite reports in Italy that a deal has been done, it is not certain the mooted loan move will be concluded.
Like Darmian and Smalling, Rojo has played no part in United’s opening three Premier League games.
Solskjaer has seven central defenders in his squad, although Eric Bailly is out until the end of December after having knee surgery.
Shaw was substituted after suffering a hamstring injury. Martial reported a thigh problem after the game.