The 26-year-old was informed by the La Liga that he was free to move on after failing to become a mainstay of Diego Simeone's setup in their triumphant 2020-21 season.
City, Liverpool and Manchester United were all linked with moves for Saul, but Chelsea ultimately secured the midfielder's signature at the last minute on an initial season-long loan.
Pep Guardiola admitted last month that Silva wanted to leave, but the report adds that City were holding out for a sum of £40m for the Portuguese, while Guardiola was not interested in welcoming Saul to Manchester.
Chelsea have the option to make Saul's transfer permanent for around £30m next summer, and the Spain international could make his debut in next Saturday's clash with Aston Villa. body check tags ::