The Eagles finished rock bottom of Group C after accumulating only two points from their three matches, losing to England before playing out draws with Slovenia and Denmark.
Milinkovic-Savic, 27, the younger brother to ex-Lazio midfielder Sergej Milinkovic-Savic, 29, was Serbia's No.1 goalkeeper at the 2022 World Cup, but he did not play a single minute at Euro 2024.
Predrag Rajkovic instead played between the sticks in all three group-stage games for Serbia, while Milinkovic-Savic acted as a backup along with Djordje Petrovic.
Milinkovic-Savic has been at the centre of an off-field altercation, though, as Serbian newspaper Telegraf claim the shot-stopper was involved in a "fight" with a Serbian fan during a day off ahead of Tuesday's final group game with Denmark.
The report claims that a Serbian fan was in disbelief as to why Milinkovic-Savic was in the bar drinking and decided to angrily confront the goalkeeper, who was accused of swearing and throwing a punch, causing bar staff to step in and split up the fight.
A separate account of events from the Telegraf report came from the bar owner, who did not want his identity to be revealed.
The bar owner is quoted as saying: "Vanja came to our bar, but the security at the entrance didn't recognise him and they didn't want to let him in because he was wearing the national team shorts, which made him angry and there was a fight.
"I soon intervened, I grabbed his hand and told him to come in with me, but he refused. Everything happened in the bar's garden.
"I didn't hear what they said at him and what they said to each other. I'm glad that in the end we resolved everything."
The Serbia Football Association are yet to comment on the matter, while manager Dragan Stojkovic and the rest of his squad have now departed Germany following their elimination from Euro 2024.
Milinkovic-Savic, a 19-cap Serbia international, plays his club football at Serie A side Torino and kept 18 clean sheets in 36 top-flight appearances last season to help the Turin-based side finish ninth in the table.