The Lazio midfielder has been a long-time target of the Red Devils, with Jose Mourinho’s men chasing the player since last summer.
The Serie A club were previously against selling Milinkovic-Savic to any European suitor.
But the club have relaxed their stance in recent times, and Chelsea are looking to take advantage.
According to Italian outlet Corriere dello Sport, the Blues have expressed an interest in the 23-year-old since Maurizio Sarri’s appointment.
Lazio are looking to rake in around €130m (£115m) for Milinkovic-Savic.
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However, Corriere dello Sport report Chelsea may try secure a cash-plus-player deal.
Cesc Fabregas could be one player to depart Stamford Bridge this summer as Sarri makes wholesale changes.
Real Madrid and Barcelona have also been linked with Milinkovic-Savic in recent months.
However, La Repubblica report Lazio are expecting the latter to end their interest as they turn their attention to Paul Pogba.
The publication goes on to claim Lazio chiefs are braced for United to launch a bid should the Frenchman leave this summer.
Milinkovic-Savic is currently enjoying a holiday after playing in the World Cup for Serbia, who crashed out at the group stage.