Nicola Balice of Calciomercato.com reported the two clubs will meet next week, and Milinkovic-Savic is expected to feature highly in their discussions.
The 23-year-old has emerged as Serie A's most in-demand talent this season, but the Red Devils may struggle to convince him of a switch to the Premier League if Juventus make enough early progress.
It was reported earlier in the year Milinkovic-Savic could be sold for €70 million (£61.5 million), but SportMediaset recently reported Lazio informed his suitors he'll cost more than double that (h/t Sport Witness):
Lazio have benefited hugely from his evolution into a box-to-box superstar this term, in particular, displaying the kind of tenacity and consistency many at United may have hoped to see from record signing Paul Pogba.
Because Pogba has at times underwhelmed in his last two seasons at United, there remains some indecision regarding manager Jose Mourinho's midfield, to which Milinkovic-Savic could be a fine addition.
The trouble could come in attempting to fit Pogba and Milinkovic-Savic into the same midfield core, as both share a host of similar attributes and might limit the team if it would inhibit either's ability to shine.
Italian football writer Adam Digby recommended Juventus go to great lengths to lure their target as a priority signing this summer:
The Bianconeri have formed a habit of cherry-picking the best talents in Serie A in a similar fashion that Bayern Munich do in the Bundesliga. Gonzalo Higuain, Miralem Pjanic and Paulo Dybala all joined from domestic rivals in recent years.
United have been one of the most feverishly speculated clubs associated with a move for Milinkovic-Savic, but the Eagles star has been professional in maintaining his resolve, per Yahoo Sport UK:
Juve can offer a boost in title prospects both foreign and domestic with a relatively short move to Turin, while the Premier League transition that comes along with undoubtedly bigger wages in England is a risk for any talent.
It seems Mourinho's side will be making up for lost time if they're to pursue a deal for Milinkovic-Savic this summer, as Juventus look to begin charting their course to what would be another Serie A coup.