Martínez spent 10 years at Arsenal, initially serving as an understudy and being loaned to various clubs before breaking into the starting line-up in 2019.

Martínez transferred to Aston Villa in a deal worth up to £20 million, signing a four-year contract.

Martínez made his debut for Villa, saving a penalty from John Lundstram in a 1–0 home win against Sheffield United.

Martínez equalled Brad Friedel's club record for clean sheets in a Premier League season, with 15, and was named Aston Villa Supporters' Player of the Season.

Martínez signed a three-year contract extension with Aston Villa, extending his stay until the end of the 2026–27 season.

Martínez was named one of the two Aston Villa vice-captains, alongside Diego Carlos.

Martínez made his 100th Premier League appearance for Aston Villa in a 2–0 away victory over Chelsea, breaking the club record for most clean sheets in a goalkeeper's first 100 games.

Martínez saved two penalties in a 4–3 victory over Lille following a 3–3 draw on aggregate, qualifying his club to the Conference League semi-finals.

Martínez signed a new long-term contract with Aston Villa, lasting until 2029.