The 18-year-old was heavily linked with a move to Old Trafford over the summer, with reports suggesting a deal was close to being agreed.
However, United spent their money elsewhere, bringing in Donny van de Beek, Facundo Pellistri and Edinson Cavani in attacking positions.
The Red Devils are thought to still hold an interest in Stevanovic but Manchester City are now close to agreeing a deal for the teenager, worth just £6m, reports Serbian site Novosti.
The deal has not been finalised, but it will go through in January if it is completed and the young winger is likely to remain at Partizan for the rest of the season on loan where he is playing regular football.
The 18-year-old has already racked up 54 appearances for the club, scoring 12 goals and has been capped at Under-19 and Under-21 level by Serbia.
The 2020/21 Serbian Super Liga started in August and he has started 10 of the 12 games so far, scoring three goals.
Stevanovic is mainly deployed as a left winger but can also operate on the opposite flank or behind the striker.