United saw an £80million move for Milinkovic-Savic knocked back by Lazio last summer, but Mourinho wants to revive transfer talks when the winter window opens.
Mourinho watched Milinkovic-Savic in action for Serbia on Thursday night, with the midfielder admitting the United boss attention was flattering.
The Sun claim Lazio are seeking a fee upwards of the £89m United paid for Paul Pogba.
Milinkovic-Savic agreed a new deal with Lazio this month, but Mourinho believes his side hold a key advantage over their rivals.
Former Chelsea player Mateja Kezman is Milinkovic-Savics agent and Mourinho is banking on his relationsip with the former Blue to seal the deal.
Mourinho signed Kezman during his first spell as Chelsea manager and United believe they are now in pole position to land Milinkovic-Savic.
The 23-year-old is viewed as the ideal replacement for Nemanja Matic, who is reaching the end of his peak years.
Milinkovic-Savic is an international team-mate of Matics and Mourinho is desperate to bring fresh blood into his squad.
Mourinho feels he was not backed in the summer transfer window, hitting out at the clubs chiefs for their lack of support in player recruitment.
United have endured a dismal start to the season having already registered three defeats in the Premier League.
Mourinho is under pressure to turn things around, and quickly, but is eager to spend big in January to change his sides fortunes.