In his early years, Barrenechea played for Newell's Old Boys.[2] In the summer of 2019, he moved to Sion for €3.1 million.[3]
Juventus
In the winter of 2020, Barranechea moved to Juventus U23 – the reserve team of Juventus.[3] He made his debut with Juventus U23 on 25 October 2020, in an away win against Lucchese.[4] On 22 May 2021, Barrenechea injured his ACL.[5] On 20 February 2022, Barrenechea returned from his injury coming on as a substitute in the 59th minute against Seregno[6] in a 2–2 draw.[7] On 2 April, Barrenechea scored the first goal in his career in a 1–1 draw against Virtus Verona.[8] On 21 May, Barrenechea scored a 12th-minute decider in the second leg of the second round of the play-offs - a 1–0 win against Padova, which did not prevent Juventus from being eliminated from the play-offs.[9]
Barrenechea started to get first-team call-ups in the 2022–23 season. He made his debut for Juventus on 2 November 2022 as an 88th-minute substitute for Juan Cuadrado in a 2–1 UEFA Champions League loss against Paris Saint-Germain.[10] By mid-February 2023, he was moved to the senior squad on a permanent basis. He made his Serie A debut for Juventus on 28 February 2023 against Torino.[11] On 1 April, Juventus announced his definitive promotion to the first team.[12]
Loan to Frosinone
On 15 August 2023, Barrenechea was sent on loan to newly-promoted Serie A club Frosinone for the 2023–24 season.[13] He made his debut for the club four days later, coming on as a late substitute for Abdou Harroui in a 3–1 league defeat to Napoli.[14] On 19 December, he scored his first goal for Frosinone in a 4–0 Coppa Italia win over Napoli.[15][16]
Aston Villa
On 1 July 2024, Barrenechea signed for Premier League club Aston Villa on a permanent deal, as part of a double-transfer from Juventus, alongside Samuel Iling-Junior.[17][18][19] The deal commanded an €8 million fee, plus €3 million in add-ons,[17][19] and was related to the transfer of Douglas Luiz to Juventus, despite being a separate transaction.[17][20]
On 30 August 2024, Barrenechea signed for La Liga club Valencia on a season-long loan.[21]