The Euro 2024 winner has undergone testing and consultations with doctors in Barcelona since the injury.
While City haven't commented on the timescale, they admitted Rodri is a long-term absentee.
A City statement read: "Manchester City FC can confirm that Rodri has suffered a ligament injury to his right knee. The injury was sustained during the first half of this weekend’s 2-2 draw with Arsenal in the Premier League.
"The midfielder traveled to Spain to seek specialist consultation this week, following initial tests in Manchester. Assessment remains ongoing to ascertain the full extent of the injury and expected prognosis.
"Everyone at the club wishes him a speedy recovery and mancity.com will bring you regular updates regarding his rehabilitation."