According to both Italian outlet Il Messaggero and Spanish newspaper AS (h/t Richard Fay of the Manchester Evening News), Real are set to make a €150 million (£132 million) bid for the midfielder, which could rise to €170 million (£150 million) with add-ons.
United are less keen to splash so much cash on the star, so Los Blancos "are believed to have the upper hand in any pursuit," per Fay.
The Lazio man, who is set to play a key role for Serbia at the World Cup, would be a significant upgrade on the Belgian.
Fellaini's height and physicality are his biggest strengths, and United boss Jose Mourinho values such qualities in his players.
As beIN Sports' Matteo Bonetti noted, Milinkovic-Savic possesses both of those traits and so much more:
Matteo Bonetti @TheCalcioGuy
The son of two professional athletes, Sergei Milinkovic might be Serie A's most dominant midfielder. At 6'3 (193cm), he has incredible balance and agility. Strong in the air, technical, creative & and eye for goal (12 goal season). Whoever lands him will have an absolute gem. https://t.co/iBoBFooxZK
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.
He looks to get the ball forward and can accurately carve open defences almost as well as anyone in Serie A, per WhoScored.com:
What's more, while he's not a holding player, he works hard defensively for the team, per Squawka Football:
Given the heights he can still go on to reach, it's hardly surprising he has caught the eye of Madrid.
Los Blancos have the financial muscle to rival United, and they'll hold plenty of appeal having won four of the last five UEFA Champions League titles, though the Red Devils would likely be able to offer him a greater role.
He'd enjoy star status at Old Trafford, whereas at the Santiago Bernabeu, he'll have to contend with Casemiro, Luka Modric and Toni Kroos for game time.
That didn't work out so well for Dani Ceballos last season as he started just nine matches, five of which were in the Copa del Rey.
Milinkovic-Savic is a year or two ahead of the Spaniard in terms of his development, but unless Real's next manager is more willing to rotate than Zinedine Zidane was in his final season, it could give him pause about going there.