Born in Sremska Mitrovica, Bogosavac joined the youth academy of Partizan from Sirmium in 2008. He signed his first professional contract with the club on 14 August 2013, alongside Miladin Stevanović, penning a three-year deal.[1] In order to gain experience, the duo was immediately assigned to affiliated team Teleoptik. They were both promoted back to Partizan in the 2015–16 campaign, with Bogosavac playing regularly under manager Ivan Tomić in the second half of the season.
On 14 February 2020, he joined Russian Premier League club Akhmat Grozny on loan until the end of the 2019–20 season, with an option to purchase.[3] On 3 July 2020, Akhmat activated their purchase option and signed a four-year contract with Bogosavac.[4]