The Lazio midfielder has been linked with a move away from the Biancocelesti, particularly since they slipped out of the Champions League places on the final day of the Serie A season.
However, there is no release clause in his contract and club President Claudio Lotito is prepared to play hardball over the Serbian's future, claiming that he has already turned down a proposal in excess of €100m.
It is believed the team behind that approach was either Juve or United, both of whom have been admirers of Milinkovic-Savic for a while. But Real Madrid could usurp both with a nine-figure bid for the Serbian.
According to the newspapers Il Messaggero and AS (via Football Italia), Los Blancos will launch a €150m offer for Milinkovic-Savic, with another €20m in performance-related bonuses on top of that.
Sergej Milinkovic-Savic, who is a Real Madrid fan & born in Spain: "That comparison (to Zidane) can only flatter me. However, I see myself more as a Yaya Toure. As for idols, I can say I have always watched and looked up to Nemanja Matic as well."— M•A•J (@Ultra_Suristic) June 9, 2018
That would take the total fee to €170m, smashing the club's previous transfer record of €100m which they spent on Gareth Bale in 2013.
Such a fee would also make Milinkovic-Savic the most expensive European footballer of all time and the second-most expensive overall, with only Neymar's €222m move to Paris Saint-Germain ahead of him.
The 23-year-old would also command a yearly salary of €6m to begin with, which could increase over the course of a five-year contract.
Milinkovic-Savić came to Utd he would easily bench Herrera, McTominay and Matic. Him, Fred and Pogba are the dream midfield.— James 🔰 (@mufcszn) June 9, 2018
He has been included in Mladen Kristajic's Serbia squad for the World Cup and has indicated that he will not make a decision on his future until after the tournament.