Mourinho was eager on bolstering his defensive department in the previous transfer window by signing a centre-back.
The Red Devils were linked with moves for Harry Maguire, Toby Alderweireld, Yerry Mina and Jerome Boateng, but eventually landed none.
Yahoo Sports also claimed that Bonucci was shortlisted as a possible target, with the player rumoured to have been unhappy at the San Siro just a year after he moved to AC Milan from Juventus in a £37million deal.
Now, a report from The Sun states Mourinho made a request to the United board that they try and sign Bonucci to add some experience to the back-line.