Richards suffered the setback when on Olympic duty for Great Britain in the quarter-final clash with South Korea.
City initially thought Richards would only be out of action for a week but the prognosis is worse than first feared.
Speaking after Sunday's 3-2 win over Chelsea at Villa Park, Mancini said: "With Micah, we thought it was a small injury but instead he will longer. He has a problem with his ankle, a big problem maybe, and it may be one month out."
Goalkeeper Joe Hart was also ruled out with a back problem which has also meant he is not available for England's friendly with Italy on Wednesday.
Mancini said: "Joe had a problem yesterday (Saturday) with his back but I don't think it is serious."