The 28-year-old limped off during City's 2-2 draw against Arsenal in the Premier League on Sunday after going down holding his right knee.
ESPN and Spanish sports daily Marca reported that tests on Monday have confirmed an ACL tear.
Rodri's loss would be a huge blow to City's hopes of winning a fifth consecutive Premier League title and ambitions of reclaiming the Champions League.
The former Atletico Madrid player has been touted as a potential Ballon d'Or winner this year after his influential role in City's latest Premier League title and Spain's success at Euro 2024.
"Rodri is strong. He leaves the pitch in this action because he felt something, otherwise Rodri stays there," City boss Pep Guardiola said after the Arsenal match.
"He's the best holding midfielder in the world, he is a potential Ballon d'Or winner. I would love for him to win (the Ballon d'Or)."
