Sarri discussed the Serbian's future following Monday night's 2-2 draw with Juventus at the Allianz Stadium when Milinkovic-Savic was again decisive for the Serie A side.
The 27-year-old scored a dramatic added-time equaliser for the Biancoceleste to guarantee a top-six finish and qualification for next season's Europa League.
"Speaking to the President, I have become convinced that if Milinkovic does leave, and that will not happen easily, he will not remain in Italy," said Sarri in his post-game press conference.
Sarri's comments will alert the likes of Manchester United and Paris Saint-Germain who have been heavily linked with the 27-year-old in recent times, but would rule out Juventus who have also been keeping tabs on the midfielder.
Lazio president Claudio Lolito last week said that he had turned down a €140 million bid in the past for his prized asset but conceded that he could be tempted to sell if 'an important proposal form a world class club' was presented.
Speaking on his future, Milinkovic-Savic praised the impact Sarri has had on his season, whilst keeping his cards close to his chest.
"I've improved under Sarri. I've scored 11 goals and provided 11 assists, so I can say this has been the best season of my career. Next season, we need a few more players so we can aim for the Champions League places." the Serbia international told Sky Italia.
Milinkovic-Savic has spent seven seasons at the Stadio Olimpico after signing from Belgian side Genk in 2015.
A run of sterling performances have attracted the attention of some of Europe's top clubs for the player whose contract at Lazio runs until 2024.
Manchester United have continually been linked for the Lazio star and Erik ten Hag could kick off a new era at Old Trafford by moving for the towering midfielder.
With Nemanja Matic leaving the club at the end of the season and Paul Pogba and Jesse Lingard likely to move on when their contract expire this summer, United will need to bolster their midfield ahead of next season's assault on the Premier League.