*Club domestic league appearances and goals, correct as of 21:03, 7 July 2024 (UTC) ‡ National team caps and goals, correct as of 14:51, 2 June 2021 (UTC)
Klimowicz trained with Borussia Dortmund's academy at a young age, while his father was playing in Germany.[3] He later joined Instituto's ranks.[2][4] He appeared professionally twice at the back end of the 2016–17 season in Primera B Nacional, participating in fixtures with Juventud Unida, aged sixteen, and Nueva Chicago as they secured sixth.[2][5] Klimowicz's first senior goal arrived on 14 October 2017 during a 1–1 home draw with Aldosivi.[2][6]
On 10 May 2019, VfB Stuttgart announced the signing of Klimowicz ahead of the 2019–20 season; with the midfielder signing a five-year contract.[7] He made his bow as a late substitute in a 2. Bundesliga match with 1. FC Heidenheim on 4 August.[2] They won promotion in his first season, with his Bundesliga debut subsequently arriving on 19 September 2020 against SC Freiburg.[2] He then scored his first goal in German football on 26 September, netting in a 4–1 victory at the Opel Arena versus Mainz 05.[2][8]
Klimowicz is eligible to play for Argentina, Germany or – with the obtainment of citizenship – Poland at international level.[3][10][11] He received call-ups to train with Argentina at the U15, U17 and U20 levels.[3][12]
On 15 March 2021, Klimowicz was called up for the 2021 UEFA European Under-21 Championship group stages by Germany; he holds a German passport due to his father's career in German football.[10][13] He debuted with the German U21 team in a 3–0 win over the Hungary U21 on 25 March 2021.[14]