He was part of the Red Star Belgrade squad that won the 1990–91 European Cup, but was still too young to feature in any of their games. After fleeing the Yugoslav Wars for the Netherlands he rejected a move to Croatia Zagreb in 1999 when he played for Dutch Hoofdklasse side Babberich.[3] Pušić later became a club legend at Babberich, when he scored both goals in the 1996 Dutch Amateur Cup final against SHO in De Kuip.[4]
On 18 October 2017, he was appointed interim manager at FC Twente after his predecessor René Hake was forced to leave.[6] On 29 October 2017, Gertjan Verbeek was appointed as the new manager, after which Pušić was demoted back to his role as assistant manager. On 26 March 2018, he was promoted to the interim manager position again, as Verbeek was fired.[7] As interim head coach, Pušić failed to save FC Twente's season and finished last in the Eredivisie,[8] and were relegated to the Eerste Divisie. For 2018–19 season, Pušić was promoted to head coach.
After finishing in first place and becoming the champions of the Eerste Divisie, and securing promotion back to the Eredivisie, Pušić was sacked on 7 May 2019.[9] He was then recruited to become the assistant manager for AZ Alkmaar in May 2019.[10]
In January 2021, Feyenoord announced signing o Pušić as assistant-manager, starting in the summer of 2021. As manager Arne Slot was previously transferred by AZ Alkmaar for having discussions about a role with Feyenoord, Pušić was transferred by AZ Alkmaar also. [11]
On 24 October 2023, Pušić left Feyenoord to become the new head coach of Ukraine club Shakhtar Donetsk.[12] On 11 May 2024, he guided the club to their 15th league title in the 2023–24 season, following a 1–0 victory over second-placed Dynamo Kyiv.[13]