Italian centre-back Bonucci is one of the top names on Mourinho's shortlist of summer transfers.
After a disappointing season, Mourinho is unsure whether Phil Jones and Chris Smalling have long term futures at Old Trafford.
AC Milan signed the Italy global from Juventus last July for €42 million (£37m) but the Serie A club would be open to a transfer this summer.
The report indicates Bayern are also keen on a deal and they must persuade Milan, who are willing to negotiate a loan move with a view to a permanent switch worth £35m.
Bonucci has 77 caps for Italy.
He was heavily linked with a move to Chelsea and Manchester City last summer, but ultimately chose to remain in his home country, moving to big spending AC Milan.
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.
However, there are suggestions that the centre-back could consider leaving the Italian peninsula this time around, especially after AC Milan failed to qualify for the Champions League.