Bürki began his career in 2007 with the reserve squad of the BSC Young Boys. In 2009, he moved to FC Thun, and half a year later to FC Schaffhausen. In the summer of 2010 he went back to BSC Young Boys, and again half a year later he was transferred to Grasshopper Club Zürich. First, he was the back-up goalkeeper, later the first goalkeeper. He was on loan until 2013, when Grasshoppers bought his rights.
SC Freiburg
On 24 May 2014, he signed a contract with SC Freiburg.[3]
On 14 June 2015, he signed for Borussia Dortmund.[5] He made his formal debut for the team on 15 August 2015, in a 4–0 win over Borussia Mönchengladbach.[6] He kept 12 clean sheets for Dortmund in the league, making 33 appearances.[7] In the 2016–17 season, he made 27 league appearances, keeping nine clean sheets.[8]
In the 2017–18 season, Bürki became the first goalkeeper to keep five clean sheets in the first five Bundesliga games.[9]
St. Louis City SC
On 16 March 2022, Bürki agreed to a deal that would send him to Major League Soccer expansion franchise St. Louis City SC for their inaugural season in 2023.[10] He is the highest-paid goalkeeper in the league and would earn a $1.5 million base salary.[11] Bürki played four matches for reserve squad St. Louis City 2 in 2022 along with several other future St. Louis players.[12]
On 20 February 2023, Bürki was named the club's captain.[13]
In 2014, Bürki was first called up for Switzerland and was also in the squad for the 2014 World Cup in Brazil. On 18 November 2014, he made his debut in a friendly match against Poland.[14] He was part of the squad for the 2016 European Championships, but never made it to any match.[15] Ultimately, his team was eliminated in the round of 16 away after losing 5–4 on penalties against Poland.[16]
He was included in Switzerland's 23-man squad for the 2018 World Cup.[17]
In 2018, Burki stated that he did not want to be called up to the Switzerland squad in order to focus on the 2018–19 season at Dortmund.[18]