Greenwood remains suspended by United amid an internal investigation after charges of attempted rape, engaging in controlling and coercive behaviour and assault against him were dropped in February.
The 21-year-old's future remains uncertain as United coach Erik ten Hag is reported to be against the club selling him, meaning he could be sent out on loan.
Roma have emerged as a possible destination for the England international, as The Sun reports Mourinho has been in touch to sound him out about making a temporary switch.
The Portuguese coach is said to have made direct contact with the player's father, while fellow Serie A sides AC Milan, Inter and Juventus are also reported to be interested.
It was Mourinho who plucked Greenwood from the youth academy and brought him on the Red Devils' pre-season tour of the United States in 2018 and he went on to sign his first professional contract that year.
However, it was under Ole Gunnar Solskjaer the following March that Greenwood would make his competitive debut for the club.
Greenwood is said to have grown "fed up" of his suspension at United, as he has trained alone, away from Carrington, in a bid to keep himself fit ahead of a possible return.
Reports in Italy have claimed that United would be happy to loan Greenwood out, and it remains to be seen if they will be receptive to Mourinho's approach.