The 19 year-old passed his medical on Sunday after Milan and Partizan agreed terms over a fee of €3.5m. The deal includes bonuses and a hefty resale percentage for Partizan.
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The Serbs had been demanding €10-15m initially, but were talked down by Milan sports chief Igli Tare this month.
Kostic made his senior Montenegro debut earlier this season and has so far scored 11 goals in 34 games for Partizan this term, mainly from off the bench.
Milan management plan to include Kostic in senior preseason training this summer before initially registering him with Milan Futuro, which will return to Serie C next season.