Born in Leioa, Biscay, Basque Country, Spain, Sainz joined Athletic Bilbao's Lezama in 2012, from lowly locals Lagun Artea. In August 2017, he signed for Alavés after refusing a contract renewal from the Lions.^[2]

On 2 September 2018, Sainz made his senior debut with the reserves at the age of just 17, playing the last three minutes in a 3–0 Tercera División home win against San Pedro. He scored his first senior goal on 7 October in a 2–0 home defeat of Beasain,^[3] and finished the campaign with five goals in 18 appearances.

Sainz made his first team – and La Liga – debut on 25 August 2019, coming on as a second-half substitute for Luis Rioja in a 0–0 home draw against Espanyol;^[4] by doing so, he became the first player born in the 21st century to appear for the senior side.^[5] He scored his first goal in the senior category on 18 June of the following year, netting the opener in a 2–0 away win against Real Sociedad.

On 5 August 2021, Sainz was loaned to Segunda División side Real Zaragoza for the season.^[6]

On 28 July 2022, Sainz moved abroad for the first time in his career after signing for Giresunspor in Turkey.^[7]

On 30 June 2023, Sainz signed for Championship club Norwich City on a three-year contract following his departure from Giresunspor.^[8][9] Sainz scored on his debut for Norwich in an EFL Cup tie against Fulham on 27 September 2023.^[10] Sainz also scored for Norwich in an FA Cup tie against Liverpool on 28 January 2024.^[11]

Following a strong start to the 2024–25 season, scoring four goals in three matches, including a hat-trick against Derby County, he was named EFL Championship Player of the Month for September 2024.^[12] He won the award for a second consecutive month following a further four goals and two assists.^[13] Sainz scored his second hat-trick of the season on 26 November with three goals against Plymouth Argyle at Carrow Road.^[14] Following an incident in a fixture against Sunderland on 21 December 2024, Sainz was charged by the Football Association in January 2025 of "spitting at an opponent",^[15] subsequently receiving a six-match ban and being fined £12,000.^[16] At the EFL awards for the 2024‍–‍25 season, Sainz was named in the EFL team of the season.^[17]

On 13 July 2025, Sainz joined Primeira Liga side Porto on a five-year contract, for a reported transfer fee of £14.25m.^[18][19] He made his debut for the club on 11 August, starting in a 3–0 league win over Vitória de Guimarães at the Estádio do Dragão.^[20] Two weeks later, he netted a brace in a 4–0 home victory against Casa Pia.^[21] On 25 September 2025, Sainz made his continental debut, starting in a 1–0 win away at Red Bull Salzburg, in the league phase of the UEFA Europa League.^[22]