Mourinho is understood to be in the market for a new central defender as both Phil Jones and Eric Bailly have struggled with injuries this season.
Tuttosport reports that Mourinho has identified Bonucci as a target to solve his defensive issues and has made contact with the 31-year-old about joining Old Trafford.
AC Milan signed the Italy international from Juventus last July for €42 million (£37m) but the Serie A club would be open to a transfer this summer.
The report claims that AC Milan would want at least €40m (£35m) for Bonucci, although they could also be open to a loan deal with United having an obligation to buy at the end of next season.
AC Milan spent over £210m on new signings last summer but having missed out on the Champions League once again, the club need to offload some of their key players.
Bonucci was heavily linked with a move to the Premier League last summer with both Manchester City and Chelsea interested.
But the Italian defender snubbed a switch to England and joined AC Milan in a surprise deal.
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