Richards suffered the injury in Team GB's defeat to South Korea during the Olympic Games, and after Richards was an absentee for his club's 3-2 win over Chelsea in the Community Shield, Mancini revealed that the 24-year-old's injury is worse than initially feared.
Speaking after Manchester City's victory, Mancini said: "We thought it was a small injury but instead it will be longer.
"He has a problem with his ankle, a big problem maybe. It may be one month out."
Richards made 29 league appearances for the club last season.