The defender's future at Old Trafford is in doubt following a difficult two years at the club.
The Italy international was signed for £12.7million from Torino in 2015 by Louis van Gaal as a long-term solution at right-back.
But Darmian has failed to establish himself as a regular member of Manchester United's first XI.
Matteo Darmian is keen on a return to his homeland in Italy
United value Darmian at a little under £22m
Under Jose Mourinho, the versatile defender has featured sparingly at left-back and right-back.
Recent reports have suggested United value Darmian at a little under £22m.
And if United look to sell at the end of the season, the 27-year-old has a preference.
Matteo Darmian has made just 11 Premier League appearances for Manchester United this season
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Tue, March 28, 2017
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John Terry – 100/1
Manchester United could turn to Celtic's Kieran Tierney to replace Matteo Darmian
According to the Daily Mail, Darmian would prefer to rejoin Serie A.
Roma and Inter Milan are said to be interested in signing the United man, but few will be ready to meet his supposed asking price.
Mourinho will need to replace the full-back should he go, and Celtic youngster Kieran Tierney is being watched repeatedly.