The Red Devils were linked with signing the Serbia international last summer but a move never came to fruition.
Ole Gunnar Solskjaer is desperate to bring in midfield reinforcements this summer as he looks to bolster his squad.
Ander Herrera and Marouane Fellaini have departed and with rumours about Paul Pogba leaving before the end of the transfer window, doing so could be even more important.
Milinkovic-Savic is highly-rated in Serie A and has been linked with an exit from Lazio for a few seasons.
And according to Italian publication Gazzetta dello Sport, he could now be set to leave Rome.
The report claims that a deal is all but agreed after Manchester United submitted a fee that the Italian side agreed.
A fee of £67m up front with another £4m in easily achievable add-ons is the price reported for Milinkovic-Savic.
The 24-year-old is said to have already agreed terms on a five-year contract worth over £100,000-a-week.
Should the deal go through in the coming days it could open up a transfer merry-go—round, starting with Pogba.
The France international has been linked with a return to Juventus or Real Madrid and an exit could be then be on the cards.
Bringing in Milinkovic-Savic could then see United cool their interest in Sporting Lisbon playmaker Bruno Fernandes.
Solskjaer is also keen to bring in the Portugal international but completing deals for both midfielders would be unlikely.
United have so far signed Aaron Wan-Bissaka and Daniel James this summer but are keen to add more new faces.
A new centre-half is still the top priority with Leicester City star Harry Maguire top of the wish-list.
Solskjaer insists the club are being patient in their pursuit of key targets, however.
“I’m very happy with Wan-Bissaka and James but we’ve got be patient,” he said. “It’s a long haul and it’s very important that we get the right ones in.
“We can’t just jump on a different path when you hit a hurdle. I’ve not had a frustrating feeling at all.