Born in Madrid, Sáiz arrived in Real Madrid's youth system at the age of eight. He made his senior debuts late into 2008–09 – one game – as the B-team competed in Segunda División B; in the following season, spent in the same level, he contributed more solidly, but they could only rank in eighth position, thus failing to reach the promotion playoff stage. For the 2010–11 campaign, Sáiz was loaned to Sevilla FC's reserves also in division three, in a season-long move. In the following transfer window, however, he returned to Real Madrid, being assigned to its C-side in Tercera División.
Melilla / Getafe
On 17 August 2011, Sáiz returned to the third tier and joined UD Melilla, with Real retaining a buyback option on the player. Late into the following transfer window, however, he terminated his contract and signed with Getafe CF's reserves in the same tier; he made his first-team – and La Liga – debut on 1 April 2012, coming on as a substitute for Jaime Gavilán for the last 12 minutes of a 0–3 away loss against Atlético Madrid.
On 5 August 2015, Sáiz joined SD Huesca from Segunda División in a season-long move. He scored his first professional goal 17 days later, but in a 2–3 home defeat to Deportivo Alavés; he netted a further two times in his first year, from 30 overall appearances. In late March 2016, Sáiz signed a permanent contract with the Aragonese club until June 2018. In the following campaign he scored a career-best 12 goals, as his team was ousted in the promotion play-offs.
Leeds United
On 10 July 2017, Huesca and Leeds United reached an agreement for the transfer of Sáiz. Three days later, the latter club announced the signing of a four-year contract after a successful medical. He made his competitive debut on 9 August, scoring a hat-trick in a 4–1 victory over Port Vale in the EFL Cup– in doing so, he became their first player to achieve the feat in his first game since Carl Shutt in 1989. His first Championship appearance took place three days later, in a 0–0 home draw against Preston North End. On 19 August 2017, Sáiz opened his league scoring account, contributing to the 2–0 win at Sunderland. The following 7 January, he was involved in an off-the-ball incident and was shown a red card for allegedly spitting at an opposition player during the 1–2 defeat to Newport County in the FA Cup; he returned from his six-match suspension on 21 February in a 2–2 draw against Derby County, providing an assist for Ezgjan Alioski. After ending the season with nine goals and seven assists, Sáiz was nominated as one of four players for Leeds' Player of The Year award on 16 April 2018. For the following campaign, he was given the number 14 shirt. On 17 December 2018, Sáiz was loaned to his former club Getafe for six months. He made his debut on 6 January 2019, playing 16 minutes in a 1–2 home loss against FC Barcelona. He scored his first goal in the Spanish top flight on 28 April, but in a defeat at Real Sociedad by the same scoreline.
Girona
On 18 July 2019, Sáiz joined Girona FC for an undisclosed fee.
