[4] On 2 December 2012, Živković came on as an 84th-minute substitute in the first team's 2–0 win over Heracles Almelo,[5] making him the youngest debutant in the history of the club.

[7] His talent did not go unnoticed, with 11 top football clubs enquiring about him by November 2013,[8][9] including the likes of Real Madrid and FC Barcelona.

[12] On 5 June 2014, Živković made his debut for Ajax in a friendly match against Wacker Innsbruck concluding the club's pre-season training camp in Austria.

[13] He made his competitive debut for the reserves team Jong Ajax in the Eerste Divisie season opener, a North Holland derby match which resulted in a 3–0 win at home against SC Telstar.

[16] On 28 October 2014, Živković made his regular season debut in the first team, playing in the third round KNVB Cup away match against SV Urk.

[25] On 11 July 2023, Živković joined Singapore Premier League outfit, Lion City Sailors on a one-year contract.

He donned the no.34 jersey in tribute to his former Ajax teammate, Abdelhak Nouri who still suffered from a cardiac arrhythmia back in 2017 and in still recovering in the process.

On 25 September 2023, he scored first club hat-trick and providing two assist in the 2023 Singapore Cup fixtures against Balestier Khalsa in a 7–0 home win.

On 4 October 2023, Živković scored his first AFC Champions League goal in a 2–1 away win against Hong Kong side, Kitchee.

On 8 November 2023, Živković scored a brace against Korean giant, Jeonbuk Hyundai Motors helping the club to another historic win.