The Serbian midfielder, 25, has been on the radar of various Premier League clubs for some time but Lazio have always maintained he is not for sale.
But now, for the first time, sporting director Igli Tare has admitted that it is up to Milinkovic-Savic if he wants to leave the Stadio Olimpico.
Tare was asked by Sky Sport Italia what would happen if Lazio received their asking price for him.
He said: 'The player's desire is crucial. Generally speaking, if a player thinks that their time is up and wants to go somewhere else, it shouldn't be a problem.
'Eras come to an end. Just look what happened to Napoli this season. The fact he stayed in Rome for five years is because he has strong links with the city and club.'
Tare added: 'When there was some temptation, a bit on our part, a bit on his, we never went all the way. However, things must be assessed at the end of the season.'
Tare's remarks countenancing a possible sale of Milinkovic-Savic represent a shift from Lazio's usual 'not for sale' stance.
Tottenham manager Jose Mourinho has admired the player for some time and wanted to bring him to Manchester United in 2018 when he was in charge there.
The Serb, who goes by the nickname 'sergeant' because of his powerful physique, leadership and strength in the tackle, could be seen as Christian Eriksen's replacement at Spurs.
Chelsea manager Frank Lampard is also a fan of the player, while Ole Gunnar Solskjaer at Man United may consider him a successor to Paul Pogba if the Frenchman departs.
Milinkovic-Savic has been linked with fellow Serie A clubs Juventus and Inter Milan but the player would not want to join one of Lazio's direct rivals.
Signed from Belgian club Genk for £16.2m back in 2015, Lazio stand to make a handsome profit if they sell a player who has scored 36 goals and contributed 26 assists in 194 matches for them.
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