The Sun is reporting that the Italian defender has asked the Portuguese coach to let him leave Old Trafford and move to Turin ahead of next season.
It is claimed that the Italian champions see Darmian as the perfect replacement for Dani Alves, who is thought to have had his contract with Juve cancelled.
The same story claims that Darmian has phoned Mourinho and informed him of his desire to move to Juventus.
The article also states that a move for Darmian to Juve would start of as a loan deal, with the option to make it permanent at the end of next season.
Manchester United have already bolstered their defensive options this summer with the signing of centre-half Victor Lindelof from Benfica.
Darmian helped Manchester United secure a return to the Champions League for next season by beating Ajax in the Europa League final last month.
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