The Daily Mail says Richards is not prepared to sign a new, five-year deal.
He has struggled for games under Manuel Pellegrini, yet the club are keen to keep him and the right back has been offered an increase on his £80,000-a-week deal to stay.
The 25-year-old's contract does not expire until the end of next season, but City may look to sell him in a cut-price deal this summer to avoid losing him for nothing in 2015. Liverpool, Arsenal and AC Milan are monitoring the situation.
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