The 30-year-old's future at the Emirates Stadium is in doubt as the French international's existing contract is set to expire in the summer and a new agreement is yet to have been reached.
Paris Saint-Germain are understood to be keeping tabs on the situation and a lucrative offer from the big spending French outfit could well appeal to him.
Gunners boss Arsene Wenger had hoped Carl Jenkinson would be Sagna's natural successor, but privately the coaching staff have concerns about the progress of the former Charlton youngster.
Jenkinson, who made his senior England debut in 2012, has struggled for form when called upon this season and was at fault for Chelsea's opening goal in the recent Capital One Cup defeat to the Blues.
Wenger has been an admirer of Richards for some time, and is ready to test City's resolve to hang on to him with an offer in the new year after watching the 25-year-old slip down the pecking order.
Pablo Zabaleta has made the right-back berth his own at the Etihad Stadium, but even injury problems at centre-back have not opened the door to the versatile Richards under new City boss Manuel Pellegrini.
Richards may therefore be willing to move on in January when the transfer window reopens, and with just 18 months remaining on his contract, City are unlikely to stand in his way unless Pellegrini's stance towards him changes in the next couple of months.
He did however, win a recall to the starting line-up for Sunday's 1-0 defeat to Sunderland at the Stadium of Light.