The Red Devils initially earmarked him as a replacement for Marouane Fellaini but, when the Belgium global stayed put, they eased off.
The Spain-born ace played for Serbia at this summer's World Cup in Russian Federation, and has been attracting interest from some of the biggest clubs around having impressed with his size and attacking ability from midfield.
23-year-old Milinkovic-Savic has been linked with leading clubs around Europe all summer but is now back in pre-season training with the Biancocelesti.
"Therefore Pogba is going to have to settle down and get on with the project at Manchester United".
But Lazio president Claudio Lotito has now told United they must spend big if they are to secure the player's signature. I have not received official offers either from [AC] Milan, Juventus, Real Madrid or Manchester United.
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And now, United are looking to rush through with a late offer to try and land the midfielder, as they look to build a squad capable of fighting rivals Manchester City for the Premier League title.
Even then, signing him will not be the easiest of tasks for Manchester United, with a number of other clubs interested in Milinkovic-Savic.
A centre back appears to be a top priority, with Toby Alderweireld, Jerome Boateng and Harry Maguire all touted, but another midfielder may yet be on Mourinho's wish list.
Ex-Genk scout Roland Janssen helped bring Milinkovic-Savic to Genk and is now working for United, with the Belgian potentially helping drive the interest in the dynamic midfielder.
"That just shows how far the kid has come".