Manchester City confirmed on Wednesday that Rodri suffered ligament damage to his right knee during their weekend clash against Arsenal, although the full extent of the injury remains unclear.
The midfielder has been in Spain this week consulting specialists after limping off midway through the first half of the match at the Etihad Stadium on Sunday after a collision with Thomas Partey.
"Manchester City FC can confirm that Rodri has suffered a ligament injury to his right knee," the Premier League champions said in a statement.
"The injury was sustained during the first half of this weekend's 2-2 draw with Arsenal in the Premier League."
The statement added: "Assessment remains ongoing to ascertain the full extent of the injury and expected prognosis."
There is no timescale on his recovery but, speaking after his team's 2-1 League Cup win against Watford on Tuesday, City boss Pep Guardiola said Rodri would be out "for a long time".
Reports earlier this week said the influential 28-year-old Spain international could miss the rest of the season.
Rodri's loss will be a huge blow to City's hopes of winning a fifth consecutive Premier League title and ambitions of reclaiming the Champions League crown.
The 28-year-old defensive midfielder played a key role as City held off Arsenal to win a fourth straight Premier League title last season and was named player of the tournament at Euro 2024 as Spain emerged triumphant.
The former Atletico Madrid man has lost just one match for club or country in 18 months – the FA Cup final to Manchester United – and has been touted as a potential Ballon d'Or winner.
City's Premier League win rate with their midfield lynchpin in the team is 74.1 per cent but without him it drops to 61.9 per cent.
Guardiola said he had the responsibility to find a way of coping without Rodri, who joined City in 2019.
"Rodri is irreplaceable," he said. "When a team doesn't play with the best midfielder in the world for a long, long time of course it is a big blow but my duty is to find a solution so that we remain competitive as we have been for many years.
"When one player is irreplaceable you have to do it as a team and this is going to happen."
Rodri had already had a frustrating start to the season, missing City's first four games, including the Community Shield, after suffering a hamstring injury.