The Serbia international has been followed by United scouts this season, but Gazzetta Dello Sport says Mourinho has stepped up their interest this month.
Mourinho wants to find a new midfield signing this summer and Milinkovic-Savic is under serious consideration.
Just last week, Mourinho said: "We need to sign a midfield player because we are going to lose Michael Carrick.
"You can accuse me of many, many things, but you cannot say that I lied to you. When I don't want to say things, I don't say (them). But I don't take you in other directions.
"When I told you a couple of months ago that we don't want to sign any striker or any winger and, yes, we are looking, and we are going to try to sign a midfield player, I told you that two months ago."