The star Spain international was sidelined with an injury during the champions' 2-2 draw against Arsenal in the Premier League at the Etihad Stadium on Sunday.
A statement on mancity.com read: "Assessment remains ongoing to ascertain the full extent of the injury and expected prognosis. Everyone at the club wishes him a speedy recovery."
The midfielder travelled to Spain to seek specialist consultation this week following initial tests in Manchester.
There had been reports claiming he could miss the rest of the reason but, while the problem is clearly serious, Pep Guardiola remains hopeful it does not prove that bad.
TrendingSpeaking before the club statement on Wednesday, the City manager said: "Still, we don't have the definitive. He'll be out for a long time, a while, but there is some opinions that maybe it will be less than we expect.
"Unfortunately he was injured. Right now, still, I cannot tell you [how long] because we're waiting the last phone calls from him and the doctors to know exactly what, definitely, he has and the type of surgery that he has to get."
Guardiola admitted Rodri leaves a large void, saying: "We don't want this but we will still have a good season. I trust a lot in my players, I have a duty to find a solution.
"Rodri is irreplaceable and the team won't play with the best midfielder in the world for a long, long time. Of course it's a big, big blow for us.
"But it's football, it's happened and my duty is to find a solution, to be competitive like we have been for many years."
Rodri was forced off in the first half of the draw with Arsenal after going down without contact and appeared to be in tears before leaving the field.
City rescued a draw against 10-player Arsenal with John Stones' stoppage-time goal preserving their two-point lead over the Gunners at the top of the table.
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cookies for this session only. Enable CookiesAllow Cookies OnceRodri was making his first Premier League start of the season as City managed his workload after winning Euro 2024 with Spain in the summer.
According to FIFPRO, he only had 22 rest days last season due to his commitments for club and country.
Last week, Rodri raised the possibility of players striking over the increasing demand placed on them due to the ever-expanding fixture list.
"I think we are close to that. I think if you ask any player he will say the same. It is not the opinion of Rodri or whatever. I think it's the general opinion of the players," he said.
"I don't know what's going to happen but it's something that worries us because we are the guys that suffer."
Rodri joined City in 2019 from Atletico Madrid and has made 260 appearances in all competitions, helping the club to four league titles as well as the Champions League in 2023.
' Zinny Boswell:
For all the debate and deliberation over the flashpoints in Sunday's title showdown, Rodri's injury could prove to be the incident that has the most repercussions for the rest of the season.
Rodri has arguably been City's most influential figure in the second half of Pep Guardiola's reign.
City have not lost a Premier League game that he has started since February 2023. Four of the 11 matches he has missed have resulted in defeat.
The champions now have to cope without him for a prolonged spell for the first time since he joined from Atletico Madrid in 2019.
Guardiola has options to replace Rodri, with Mateo Kovacic the player who tends to deputise at the base of midfield, but City are undoubtedly weaker without the 28-year-old.
It is no coincidence that City have won four titles in a row with Rodri an almost ever-present. He is the heartbeat of the team.
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