The 23-year-old has been one of the most sought-after commodities during this summer's transfer window, and the Premier League and La Liga titans are leading the race for his signature.
Italian newspaper Il Messaggero (via The Sun) report that the two clubs entered talks with I Biancocelesti last week, and Milinkovic-Savic's agent met with Lazio president Claudio Lotito and sporting director Igli Tare to discuss the player's future.
Considering the talent on hand (Milinkovic-Savic scored 12 goals and assisted four others in 35 league games last season), as well as the heavyweights vying for his signature, Lazio are likely to demand upwards of £120m to let the midfielder leave Italy.
The massive price tag could hand the advantage to Los Blancos, who may see a significant bump to their own coffers soon if they let Cristiano Ronaldo go to Juventus for a reported £88m. The Red Devils, on the other hand, have already spent £52m on Fred - but if the club are serious about getting their man, they have the financial backing to meet Lazio's demands.
Milinkovic-Savic played every minute of Serbia's World Cup campaign, though the Orlovi disappointingly failed to progress past the group stages.
After Serbia's exit, the 6ft2in rock claimed he needed time to recuperate before making a decision on his future, while simultaneously insisting he's happy to stay in Rome. Lazio will likely be happy with their star man's comments, but if their asking price is met, they'll likely be resigned to letting him go.