Born in Arilje, he made his first footballing steps at local club Vranić. He later joined the youth system of Partizan after being scouted by Dušan Trbojević.[1]
On 9 December 2018, Stevanović made his official senior debut for Partizan in a 3–0 away league victory over Rad, coming on as an 82nd-minute substitute for Đorđe Ivanović.[2] He made three more league appearances through the remainder of the season.
On 1 August 2019, Stevanović scored his first goal for Partizan after coming off the bench in the return leg of the 2019–20 UEFA Europa League second qualifying round, a 3–0 home win versus Welsh side Connah's Quay Nomads.[3] He thus became their youngest ever scorer in UEFA competitions and the second youngest in club history, at 16 years and 311 days old.[4] Three days later, Stevanović netted his first league goal for Partizan, opening the scoring in an eventual 4–0 home victory over Mačva Šabac.[5] He subsequently contributed with a brace in a 3–0 home win versus Rad on 18 August.[6]
In October 2020, it was announced that Stevanović had agreed to join Manchester City in the following January.[7]
On 8 August 2023, Stevanović returned to the Netherlands and joined RKC Waalwijk on a season-long loan.[8] On September 24, Stevanović scored the winning goal in a 1–0 win over FC Twente.[9]