Meanwhile, Juventus have also registered their interest in the midfielder but Lazio President Claudio Lotito has made it clear that he would rather sell Milinkovic-Savic overseas.
With Real Madrid prepared to offer more money, Savic could well be on his way to the Los Blancos. Manchester United also want to sign Milinkovic-Savic this summer and they are aware of the rivalry from the European Champions.
Fellaini's height and physicality are his biggest strengths, and United boss Jose Mourinho values such qualities in his players.
Where do you think Milinkovic-Savic will be playing next season?
What's more, after signing Fred from Shakhtar Donetsk, the Premier League outfit will only make a move for Milinkovic-Savic if Marouane Fellaini does not sign a new deal at Old Trafford.
But Milinkovic-Savic has played down any chance of a move before the World Cup, by saying he is focused on representing Serbia rather than his club future, which will be sorted after the tournament.
The 23-year-old finished the campaign with 14 goals and nine assists in all competitions, showcasing the impressive impact he can have in the final third.
Real Madrid, on the other hand, could be set to enter a period of change.
Madrid, who are currently in the process of looking for a new coach following the departure of Zinedine Zidane, recently claimed a record third consecutive Champions League title and are now thought to be looking to reshape their squad ahead of the new season.