The right-back is Louis van Gaal's top defensive target ahead of the new Premier League season with Real Madrid refusing to part ways with Sergio Ramos.
Darmian has emerged as one of the top full-backs in Europe, and is a regular in Antonio Conte's Italy squad.
Van Gaal is keen to strike a deal, but Sky Italia claims that Torino has rejected a bid worth £10.5m from the Red Devils.
It is added that the Serie A outfit want no less than £14m, although it is unclear whether United will return with an improved offer.
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Matteo Darmian has been linked with a move to Man Utd
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