Born in Belgrade, Adžić joined Red Star Belgrade at the age of 6. He joined the first team of Red Star Belgrade for the 2015–16 season, after he passed all youth categories of the same club. At the beginning of 2016, during the winter break off-season, he suffered an anterior cruciate ligament injury, and later he was out of the field until May same year. He signed his first three-year professional contract with club on 3 March 2016. Adžić returned the next season, but stayed with youth team until the end of first half-season in youth league, after which he rejoined the first team. In the meantime, he made his official debut for Red Star Belgrade in a cup match against BSK Borča, played on 26 October 2016 at the Rajko Mitić Stadium. Adžić made his Serbian SuperLiga debut in away match against Metalac Gornji Milanovac replacing Srđan Plavšić in 83 minute, and he also scored his first senior goal for 2–1 victory in last minutes of the game. Adžić scored his second goal for the team in 23rd fixture match of the 2016–17 Serbian SuperLiga season, against OFK Bačka, when he also joined the game as a substitution. Adžić started his first senior match on the field in last fixture match of the season, against Radnički Niš. During the summer 2017 pre-season, Adžić presented as an one of the club's top 10 youth prospects. He converted his squad number before the 2017–18 UEFA Europa League qualifications, taking jersey 11. On 29 June 2017, Adžić made his first continental appearance, replacing Ricardo Cavalcante Mendes during the match against Floriana. Adžić joined his first eternal derby, replacing Slavoljub Srnić in 90 minute of a match played on 27 August 2017. Adžić scored his first season realising direct free kick goal in the Serbian Cup match against Dinamo Vranje 11 October 2017. Later, during the same match, he completed his first hat-trick in professional career, by two Aleksandar Pešić's assists. In opening match of the second half- season of the 2017–18 Serbian SuperLiga campaign, played on 17 February 2018, Adžić scored his second league goal, against Javor Ivanjica. Several days later, on 21 February, Adžić also joined the in second leg against CSKA Moscow in Round of 32 match of the UEFA Europa League. On 18 March 2018, Adžić realised a direct free kick in 28 SuperLiga fixture against Mačva Šabac. Replacing Slavoljub Srnić in second half of the match against Čukarički, played on 25 April 2018, Adžić scored and made an assist to El Fardou Ben Nabouhane in 3–0 victory. In May 2018, he suffered a toe injury and missed the rest of season.
Anderlecht
In May 2018, reports of interest from Anderlecht surfaced, after which Adžić refused to sign a new contract with Red Star. On 23 May 2018, he signed a four-year contract with Anderlecht and the transfer fee was reported to be €800,000.
Emmen (loan)
Adžić scored on his debut for Emmen on 21 February 2020 while playing against Willem II.
