Bergström joined English Premier League side Chelsea from his native side TPS in 2018.[3] He made his youth debut in the final game of 2018–19 season against Reading and turned professional in September 2019. On 14 September 2021, Bergström was named among substitutes for the Champions League group match against Zenit Saint Petersburg at home.[4] In 2023, Bergström travelled to the USA as part of the Premier League Summer Series. At the start of the 2023-24 Premier league season, Marcus Bettinelli and Robert Sánchez had received injures allowing Bergström to be named on the bench on multiple occasions.[5]
Loan to Peterborough United
In July 2022, Bergström was sent on loan to Peterborough United in the League One.[6] On 30 July, He made his professional debut in a 3–2 win in a League One match against Cheltenham Town.[7] Bergström managed 9 clean sheets in 28 matches in all competitions.
Loan to IF Brommapojkarna
On 29 February 2024, Bergström extended his contract with Chelsea, and immediately thereafter IF Brommapojkarna announced that Bergström would join the team for the duration of the 2024 Allsvenskan season.[8][9] However, the loan was terminated on 18 June.[10]