The 30-year-old, who had entered the final year of his contract at the Etihad Stadium and was the subject of interest from both Barcelona and their great Spanish rivals Real Madrid, had made it clear where he wanted to go.
But Barcelona had seen an opening offer of Stg 40m ($A77 m) rejected last week and failed with an increased bid before finally reaching an agreement on Sunday (Monday AEST).
Speaking after City's 3-0 Community Shield loss to Arsenal on Sunday, Enzo Maresca called Rodri a "top player" and said "all the teams need Rodri", but City have been unable to persuade their key man to stay and sign a new contract.
Rodri had only returned to the club on Friday, having undergone back surgery following the World Cup, with Maresca saying he had given him a "big hug" when he first met up with him.
Rodri has spent seven seasons with City, winning the Premier League four times and scoring the only goal when they beat Inter Milan in the 2023 Champions League final.
The following year he won the Ballon d'Or but by the time he was presented with the trophy he was sidelined with an anterior cruciate ligament which ruled him out of most of the 2024-25 campaign.
Hamstring and groin problems also limited him to 33 appearances in Pep Guardiola's final campaign in charge.
City have already undergone major changes in midfield this summer, spending Stg 116m ($A222 m) on Elliot Anderson, while agreeing a deal to sell Tijjani Reijnders to Al Qadisah.Â
They have been linked with Lille's Ayyoub Bouaddi and Chelsea's Enzo Fernandez.
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