Internationally, Živković represented his country at every level from under-17 to under-21.
Club career
Partizan
After starting out with his hometown club Sloboda Užice, Živković joined the youth system of Partizan in 2005. He was subsequently promoted to their first team in the spring of 2007, before signing his first professional contract with the club on 1 June 2007, on a five-year deal.[1] Živković initially played for Teleoptik, before returning to Partizan. He made his competitive debut for the club in a 5–1 home league win over OFK Beograd, on 8 December 2007, coming on as a substitute for the injured Darko Božović.[2]
In the summer of 2010, Živković returned to Partizan, being projected as a backup goalkeeper to Radiša Ilić.[3] He dropped out of the 18-man squad after Vladimir Stojković joined the club in late August 2010. Eventually, Živković made one league and one cup appearance in the 2010–11 season. He was again loaned to Metalac Gornji Milanovac in the summer of 2011, making 13 league appearances in the 2011–12 season.
After leaving Gornji Milanovac for the second time, Živković started the 2012–13 campaign at Teleoptik, before being promoted back to Partizan in October 2012, as a replacement for the injured Vladimir Stojković. He recorded one league appearance until the end of the season, as the club won its sixth consecutive championship title. In the following season, Živković made his debut in UEFA competitions, coming on as a substitute for the injured Milan Lukač in the second leg of the Champions League third qualifying round versus Ludogorets Razgrad on 6 August 2013.[4] He managed to make four league appearances in the 2013–14 season.
In the fall of 2014, Živković was still the second-choice goalkeeper to Milan Lukač. He only appeared in the Serbian Cup first round, keeping a clean sheet in a 1–0 away win over Bežanija on 24 September 2014. However, Živković earned himself a place in the first eleven at the start of the spring season, quickly becoming an irreplaceable member of the team.[5] He continued to play regularly in the fall of 2015, but was removed from the squad after the arrival of new manager Ivan Tomić.[6]
Xanthi
On 10 June 2016, it was announced that Živković officially signed with Greek club Xanthi.[7] He played the full 90 minutes in all of his team's 30 appearances in the 2016–17 Superleague Greece, conceding only 25 goals in the process. In the following 2017–18 campaign, Živković remained Xanthi's first-choice goalkeeper, conceding 29 goals in 29 games.
On 21 April 2019, the Serbian goalkeeper announced via his facebook account his decision to leave the club, upon the expiration of his contract.[8]
PAOK
On 26 June 2019 , Zivkovic’s contract with Xanthi is expiring, and PAOK pounced to sign him as a free agent. The reliable 30-year-old will form the back-up for Alexandros Paschalakis after Rodrigo Rey was shifted out on loan to Pachuca. He signed a three years' contract for an €300,000 annual fee.[9] The Serb played in 52 matches, keeping 21 clean sheets, while also winning the Greek Cup of the 2020-21 season. Zivkovic has been with the club for three years, which will become five years under the new contract signed by the Serbian goalkeeper. After the departure of Paschalakis, PAOK decided to keep Zivkovic for the position of substitute goalkeeper.[10].On 7th June 2024,it was decided by the manager team of PAOK that Zivko Zivkovic's contract would not be renewed and he would depart from the club upon the expiry of his contract.