The defender was taken off the field on a stretcher after picking up a knock on Saturday.
Chasing back to cover a quick counterattack from the away side, the England international appeared to twist his ankle while trying to position himself.
Richards immediately dropped to the turf holding his knee inside the City penalty area before being given oxygen and stretchered from the pitch.
The news will come as a blow for Mancini with fellow right backs Pablo Zabaleta and Maicon also ruled out through injury.
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