Speculation was rife in Italy over the weekend that the Red Devils had won the race for the powerful 23-year-old, who has attracted interest from across Europe thanks to his impressive displays for the Serie A outfit this season.
Manchester United scouts are understood to have watched the player at least a dozen times during the campaign but senior sources have insisted no final decision has yet been made with regards making a concrete bid.
It is felt his club are trying to use United's interest to kickstart an auction, with rivals Manchester City and Bayern Munich also being linked with his signature.
Jose Mourinho wants two midfielders this summer and is also believed to be in the driving seat for Shakhtar Donetsk's Brazilian star Fred.
The money is certainly there for the United boss and the message coming out of Old Trafford is that Milinkovic-Savic would seemingly favour a switch to the Manchester giants because of his close relationship to his Serbia international colleague, Nemanja Matic, who he has tried to model his game on.