Meanwhile, Juventus have also registered their interest in the midfielder but Lazio President Claudio Lotito has made it clear that he would rather sell Milinkovic-Savic overseas.
It's likely that the midfielder will be loaned out, or perhaps even sold, this summer as Liverpool are set to be well-stocked with central midfield options ahead of next season.
Milinkovic-Savic has been closely linked with a transfer away from Lazio after they didn't safe Champions League soccer subsequent season.
Milinkovic-Savic featured 48 times over the course of the campaign, with 14 goals and 9 assists to show for his efforts.
The 23-year-old midfielder is valued by his club at around £100 million, but speculation surrounding a switch has intensified amid interest from United. Manchester United have always been keeping tabs on his situation but the Red Devils now face competition from 13-time European Champions Real Madrid.
But, Jose Mourinho has himself iterated that he will be on the lookout for two midfielders this summer if Marouane Fellaini leaves, which looks like a distinct possibility, means the Red Devils are still in the market for one more midfield addition.
Real Madrid, on the other hand, could be set to enter a period of change. With Perez looking to freshen things up in the middle of the park, Milinkovic-Savic is a viable target.
According to Il Messaggero newspaper and Spanish paper AS, the Blancos are prepared to offer €150m plus another €20m in performance-related bonuses.