The 30-year-old midfielder will move for a fee of around 75 million euros, according to reports.
"In the coming days there will be a statement and (the transfer) will become official," said the Barcelona source.
Rodri had been linked with Real Madrid after Spain won the World Cup in July, but Barcelona pushed to sign the former Atletico Madrid star, who has played a key part in Manchester City's success since he joined in 2019.
The midfielder, who won the Ballon d'Or in 2024, has struggled with injuries over the past two seasons but shone as Spain won their second World Cup.
Barca's Dutch midfielder Frenkie de Jong is set to miss several months through injury, encouraging the club to move for Rodri.
The Spanish champions had two bids rejected for Rodri but eventually reached a deal with City on Sunday, with the midfielder following Anthony Gordon, Karim Adeyemi and Jesse Bisiwu to Camp Nou this summer.
Rodri, Spain's captain, was a vital figure for Pep Guardiola at City, leading them to the 2023 Champions League and four Premier League titles among 12 major trophies.
Barca are also expected to complete a deal to sign Joao Cancelo in the coming days after they loaned him last season from Al Hilal.
As the summer transfer window ramps up, Flashscore has everything you need to keep up with breaking news, rumours and analysis. Follow our live transfer blog to follow the biggest moves, whilst expert Dean Jones will be on hand to provide the inside stories from around the world.
