Club career
Musso trained at the academy of Club de Regatas San Nicolas, before going on loan to Defensores de Belgrano de Villa Ramallo, with which he won the local championship at the age of 16. In the final, he saved three out of five penalties in the shoot-out.
Racing Club
Musso was the third-choice goalkeeper during Racing Club's 2014 Argentine Primera División championship, the team's first title in 13 years. He made his professional debut three years later, on 28 May 2017, in a 2–1 home win against San Lorenzo. Musso became the team's first-choice goalkeeper during the 2017–18 Argentine Primera División, in which Racing finished in 7th place and qualified for the 2019 Copa Sudamericana.
Udinese
In the summer of 2018, Musso was signed by Serie A side Udinese. Throughout the season, he established himself as the undisputed starter and became one of the team's key players. In three seasons, Musso accumulated a total of 102 appearances in the league.
Atalanta
On 2 July 2021, Musso was signed by Atalanta. On 22 May 2024, he clinched his first trophy with the club as Atalanta defeated Bayer Leverkusen to win the UEFA Europa League.
Atlético Madrid
On 27 August 2024, Musso joined La Liga club Atlético Madrid on a season-long loan for the 2024–25 season.
On 10 June 2025, Atlético Madrid made the transfer permanent and signed a three-season contract with Musso.
