The 17-year-old will remain with his current club next season before making the move to England.
Milinkovic made six appearances for Serbia in the qualifying phase for the UEFA Under-17 Championship, keeping three clean sheets, but they just missed out on reaching the tournament.
He would follow in the footsteps of compatriots Nemanja Vidic and Zoran Tosic if he makes his way into the United first-team in the coming years.
The deal kicks off United's transfer business for the summer, although they remain without a permanent manager in the wake of David Moyes' dismissal last month - reports suggest Louis Van Gaal will be appointed early next week.
Who makes these decisions when there is no manager in charge? How do we know that whoever takes over will wat this player at the club? Look at last summer, how Wilfred Zaha was signed before Moyes was appointed only to be shipped out on loan fairly quickly. Surely common sense would stop transfers of players while there is no manager at a club- shea666