Richards has made just two appearances under new boss Manuel Pellegrini and is said be seeking assurances about his future.
With only 18 months left on his current deal, it had been thought the Magpies could test City’s resolve with a bid in January, but it seems that they would be unable to meet the defender’s financial demands.
The exit door might not be closed, however, with Arsenal, Tottenham and Chelsea all said to be monitoring the England man’s situation.
And he may favour a move away from the Etihad Stadium in order to try and force his way in Roy Hodgson’s squad for next summer’s World Cup in Brazil.
Should a move go through, City could turn to Reece Wabara, who has impressed on loan at Doncaster Rovers, to provide cover for first-teamer Pablo Zabaleta.