Jose Mourinho wants two new midfielders for the summer and the powerful Milinkovic-Savic has been impressing for Lazio this season.
However, he struggled to make an impression last Thursday, under the scrutiny of United staff, as Lazio lost 4-1 to Red Bull Salzburg in the Europa League.
The 23-year-old has scored 11 goals this season and is valued at £89million by his club.
United need to replace Michael Carrick and Marouane Fellaini.