The 27-year-old is understood to be the Red Devils' top alternative to Frenkie de Jong, who remains keen to stay at Barcelona amid strong interest from Erik ten Hag's side.
Milinkovic-Savic was once again a productive force in the Biancocelesti engine room last term, registering 11 goals and 12 assists from 47 matches in all competitions.
The Serbia international has now totalled 58 strikes and 51 assists from 294 games for the Italian giants, winning one Coppa Italia and two Supercoppa Italiana crowns since 2015.
However, Milinkovic-Savic now only has two years left to run on his deal with Maurizio Sarri's side, and Man United are supposedly preparing to table a £42m bid for the midfielder.
The Red Devils have already lost Paul Pogba, Nemanja Matic and Andreas Pereira this summer, with the former re-joining Juventus as a free agent six years after leaving in an £89m deal.
Pogba has experienced a nightmare start to life back in Italy, though, as he suffered a serious knee injury in pre-season training, and there are fears that he could now miss the 2022 World Cup.
According to Italian outlet Correire dello Sport - via the Daily Mail - Juve boss Massimiliano Allegri has therefore identified Milinkovic-Savic as an ideal signing to replace the injured Pogba.
The report adds that Allegri also views the Serbian as an ideal partner for Pogba when the Frenchman is back fit, and Milinkovic-Savic would prefer to remain in Italy's top flight.
The midfielder's brother Vanja Milinkovic-Savic is currently on the books at Torino, and a switch to Juventus would therefore see the Lazio man move closer to his family member.
Man United are said to have made one final contract offer to De Jong, but president Joan Laporta is still expecting the Dutchman to stay put at Camp Nou.
"He [De Jong] wants to stay and we want him to stay. That is the most important. What is clear is that we have a new salary scale and all the players have to fit into it. We are trying to make these players understand the reality of the club," Laporta recently told the press.
"We find ourselves with a salary mass triggered by more than 40 per cent. We all have to make an effort. We are going to do everything possible to make him stay and we hope that the player will also do everything possible to stay."
De Jong is understood to be owed as much as £14m in deferred wages by Barcelona, which is another factor in his decision to rebuff interest from Man United.
In spite of Pogba's injury, Juventus have terminated Aaron Ramsey's contract, while Arthur Melo and Adrien Rabiot are both expected to leave in the coming weeks.
Meanwhile, Lisandro Martinez and Christian Eriksen have arrived at Man United to provide alternative midfield options, as well as new left-back Tyrell Malacia.