Born in Girona, Catalonia, López joined RCD Espanyol’s youth setup in 2007 at the age of 12, after starting out at hometown club Girona FC. He made his senior debut with the former’s B-team in the 2013–14 season, in Segunda División B.

On 12 June 2014, López signed a new four-year deal with the Pericos, being also promoted to the main squad. He played his first match as a professional on 17 December, starting in a 1–0 home win against Deportivo Alavés for the campaign’s Copa del Rey and being first-choice in the tournament as his team reached the semi-finals for the first time in nine years.

López’s debut in La Liga occurred on 1 February 2015, after starter Kiko Casilla was sent off in the 40th minute of the fixture at Sevilla FC for handling Iago Aspas’ shot outside of the penalty area, and he conceded once in an eventual 2–3 defeat. After the latter’s departure to Real Madrid in the summer, he became first-choice.

López made his international debut on 26 December 2015, having a Man of the match performance for Catalonia in a 0–1 friendly loss to the Basque Country. On 28 March 2016, he played his first game with the Spanish under-21 team, in a 1–0 win against Norway at the Estadio Nueva Condomina.

On 6 January 2016, during an away fixture against FC Barcelona for the domestic cup, López repeatedly stamped on Lionel Messi after the two players became entangled, in an action that eluded the referee but was caught on camera. On 31 August, he joined Premier League club Tottenham Hotspur on a season-long loan deal, being third-choice behind Hugo Lloris and Michel Vorm and failing to make a single competitive appearance.

On 17 May 2016, López and Espanyol teammate Marco Asensio were called up to the full side for a friendly against Bosnia and Herzegovina. He only won his first cap more than two years later, however, after coming on as a 75th-minute substitute for Kepa Arrizabalaga in a 1–0 defeat of the same opposition in Las Palmas.

On 4 July 2018, López signed a five-year contract with fellow top tier club Real Betis on a free transfer. Under Quique Setién, he missed just five league matches during his only season as the Andalusians finished in tenth position.

López joined A.S. Roma on a five-year contract on 9 July 2019, in a reported €23.5 million deal. He made his Serie A debut on 25 August, in a 3–3 home draw to Genoa CFC.