The 27-year-old has been a standout performer for the Serie A outfit during the 2021-22 season, scoring 11 goals and providing 12 assists in 47 appearances across all competitions.
Erik ten Hag will look to bolster his squad this summer and the Dutchman is expected to recruit at least one new midfielder, with Paul Pogba, Nemanja Matic, Jesse Lingard and Juan Mata all set to leave the club when their contracts expire next month.
According to La Reppublica, via LazioNews24, Man United have tabled a €55m offer for Milinkovic-Savic following a conversation with the player's agent Mateja Kezman, and further talks are set to be held within 'the next few days'.
The report adds that Lazio value the Serbia international at around €80m (£68.5m), but Kezman believes an agreement could be reached on a price between €65m (£55.7m) and €70m (£60m).
Lazio's president Claudio Lotito insists that Milinkovic-Savic is "not for sale", while head coach Maurizio Sarri believes the midfielder "will not remain in Italy" if he were to leave the Stadio Olimpico.
Milinkovic-Savic, who has been with Lazio for the last seven years, is under contract until June 2024. body check tags ::