The Serbia international has long been linked with a switch to some of Europe's top clubs, including Manchester United, but the two Milan giants are hoping to steal a march in the race to land the 24-year-old.
Milan pushed to sign Milinkovic-Savic in 2018 and will require Lazio to drop their asking price from what had been around €100m (£86m/$114m) if they are to sign the midfielder.
Gennaro Gattuso's side would also need to qualify for the 2019-20 Champions League to be able to attract Milinkovic-Savic, which, as it stands, they're currently achieving as they lie fourth place in the Serie A table.
Goal understands the player harbours doubts about moving to the Serie A champions, however, feeling that he could find himself in and out of the team in what is a supremely talented squad of players.
Indeed, Juve already possess the likes of Miralem Pjanic, Sami Khedria, Emre Can, Blaise Matuidi and Rodrigo Bentancur that play in that area of the pitch, while Aaron Ramsey will also be joining in the summer from Arsenal.
Interest from abroad has been mooted in the past, with Manchester United and Real Madrid having reportedly been in for Milinkovic-Savic in the summer.
However, his struggles to replicate his 2017-18 form, that saw him score 14 goals and register eight assists in all competitions, would suggest that some of Europe's top clubs will look elsewhere when the summer transfer window opens.
The midfielder's numbers in the current campaign are in stark contrast, with Milinkovic-Savic having scored four goals in all competitions up to now, while he has just one league assist to his name.
Lazio supporters have not been impressed with his performances in 2018-19 either, with Lazio fans, prior to a Serie A clash with Fiorentina earlier in the season, having unfurled a banner which read, "Milinkovic and Luis Alberto [are] fake talents only hunting cash!"
Lazio renewed their star man's contract until 2023 back in October, although that is unlikely to stop them selling should a big club come up with a tempting offer, especially considering his current standing with the fans.