Rodri was into the final 12 months of a contract he signed with City in 2022, and although the Premier League side wanted to give him an extension, talks did not yield a result.
Real Madrid looked best placed to sign the midfielder, but sources told BBC Sport Barcelona had Rodri’s approval to open negotiations with City.
Speaking to Spanish radio in March about the prospect of joining Real Madrid, former Atletico player Rodri said: “There are more players who have taken that path. Not directly, but over time. You can’t turn down the best clubs in the world.”
However, Real Madrid ended their interest and after City rejected Barcelona’s opening offer of £38.5m, an agreement was reached for a figure well in excess of that initial approach.
On Monday, Rodri flew to Barcelona and told journalists at the airport that it is a “dream” to play for the Catalan side.
Rodri is the third long-serving player to leave City this summer, with John Stones and Bernardo Silva both departing when their contracts expired. Silva joined Real Madrid and could face Rodri in El Clasico.
City have also sold defenders Nathan Ake and Manuel Akanji this summer, as well as goalkeeper James Trafford.
City spent £116m to sign England midfielder Elliot Anderson from Nottingham Forest, and are in advanced talks with Lille over a move for teenage Morocco midfielder Ayyoub Bouaddi.
BBC Sport understands Enzo Maresca’s side have also not given up hope of signing Chelsea’s Enzo Fernandez.