Šverko went through the youth ranks of Karlsruher SC. On 27 November 2016, the 14th matchday of the 2016–17 season, he made his debut on the first team of Karlsruher SC in a 0–0 game against 1. FC Kaiserslautern.[2]
In January 2017, Šverko moved to 1. FSV Mainz 05 on a deal until 2021.[3] On 25 February 2017, he made his debut for Mainz' reserve team in the 3. Liga in a 3–1 win against Preußen Münster.[4] After four more games for Mainz II, Šverko and the team was relegated to the Regionalliga.
For the 2018–19 season, Šverko returned to Karlsruher SC on loan.[5] At the beginning of the season he was injured and therefore lost the competition for the position of the left back against Damian Roßbach. Therefore, he made only two league appearances since his return.[6] His loan was not extended, so he returned to Mainz and played another 15 games in the first half of the 2019–20 Regionalliga season. On 21 January 2020, he was loaned out to 3. Liga team SG Sonnenhof Großaspach until the end of the season.[7]
On 31 January 2023, Šverko joined Venezia in Italian Serie B on loan until the end of the 2022–23 season.[9] On 7 July 2023, Venezia made the transfer permanent and signed a three-year contract with Šverko.[10]