The 34-year-old has struggled with injuries since joining City from West Ham in 2003.
Sinclair said: “I have enjoyed my time at City enormously and have fulfilled my boyhood dream of playing for the club I have always supported.
“I have not played as much as I would have liked recently and I am excited by the thought of a new challenge which will give me the opportunity to play more regularly.
“I feel I can continue to play in the Premier League for at least another couple of years, if not longer.”
