The 19-year-old has four caps for Montenegro and scored ten goals in the Serbian top flight last season.
Kostic could begin his Milan career with the club's second team.
Milan have also made contact with Luka Modric over extending his stay at the club for a further season.
According to La Gazzetta dello Sport, owner Gerry Cardinale personally called the Croatian to express Milan's wish to keep him beyond this summer.
Modric's World Cup with Croatia ended when his side were eliminated, but his future in club football remains open.
The option in Modric's contract for the 2026-27 season expired on 30 June, meaning any extension would require a fresh agreement.
Head coach Rúben Amorim has also called Modric in recent days, underlining the player's importance to the new manager's plans.
There is no final decision yet, with Modric set to take a few days to consider his options.
Both parties are understood to want to continue together, with Modric's daughter Ema also playing for Milan's Under-13 side.
Modric knows he could return to Real Madrid, but his priority has always been Milan, and helping the club return to the Champions League after missing out last season would fulfil a personal ambition.