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 further down City's defensive pecking order.
And 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.
‘I’ll never say never to managing my country, but I’m really happy now. As long as they’re not competing against the USA, I’ll always cheer on the Lionesses’: Emma Hayes on career future
‘Scoring five goals in nine minutes is an epic thing. Someone cleared another chance off the line and I was a bit angry afterwards – I could have had six!’: Robert Lewandowski bemoans failing to hit double hat-trick while at Bayern Munich
