The Serbia international has been heavily linked with an Old Trafford switch on the peninsula as a potential replacement for Paul Pogba.
Corriere dello Sport reports that developments are expected over the weekend as Lazio are travelling to England for a friendly match against Bournemouth on Friday evening.
The Serie A side will use the trip as an opportunity to hold talks with United officials in London, where the midfielder’s agent Mateja Kezman has already based himself for several days.
Ole Gunnar Solskjaer’s side are understood to have prepared a bid of €75 million plus a further €15 million in bonuses for the 24-year-old.
However, any move could depend on Pogba’s sale, while the Red Devils have also been linked with a swoop for Barcelona star Ivan Rakitic recently, which would rule out the signing of Milinkovic-Savic.
Lazio president Claudio Lotito has left the door open to a sale throughout the summer and again underlined that he is ready to listen to offers earlier this week.
"We'll see the conditions that line up," he said to Sky Italia.
"Milinkovic has a good rapport with the club, just as the club has with him.
"If some offers were to be presented, we'd evaluate them with a spirit of collaboration that looks after the interests of the club and the player.
"We'll make the decisions that are best for everyone. The investments we made last summer should've allowed us to qualify for the Champions League, but we hope to have learned from the experience."
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