The long-serving 26-year-old has been linked with a switch away from the Etihad Stadium throughout the transfer window after the arrival of Bacary Sagna pushed him down City's defensive pecking order.
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Richards, capped 13 times by England, will now ply his trade in Italy for the 2014-15 campaign.
A statement released on City's official website read: "Everyone at MCFC wishes him the very best for the coming campaign."
Richards made his City debut in 2005 having come through the club's academy.
He has made a total of 245 appearances for the Premier League champions.