After coming through the youth system of Partizan, Popović was loaned to affiliated club Teleoptik in the summer of 2017. He was promoted back to the Partizan first team during the 2018 winter transfer window.[1] After failing to make his official debut for the side, Popović was again loaned to Teleoptik in the summer of 2018. He was recalled back to Partizan in early 2019.
On 19 June 2020, Popović made his official debut for Partizan in a 1–0 away loss at Vojvodina in the last round of the COVID-19-shortened 2019–20 Serbian SuperLiga.[2] He would become the first-choice goalkeeper for the 2020–21 Serbian SuperLiga, mainly due to new league requirements for under-21 players, making 21 appearances in the process.
Upon the departure of Vladimir Stojković, Popović made his debut in European competitions on 29 July 2021, keeping a clean sheet and saving a penalty in a 2–0 away victory over Slovak club DAC Dunajská Streda in the UEFA Conference League.[3] He extended his contract with Partizan two months later, keeping him with the club until 2024.[4]
Aleksandar Popović officially left Partizan on July 28, 2023 and signed a contract with DAC Dunajská Streda. The young goalkeeper signed a three-year contract with the new club, while Partizan received €210,000 from the transfer, plus certain percentages from the next sale. Popović made his debut in goal on August 5, 2023 in a 5–2 win over FC Košice.[7]