The 34-year-old had made 145 appearances for Swansea since joining on a free transfer from Southend United in 2008, and helped it win promotion to the Premier League through the play-offs in 2011.
But Gower has found opportunities limited since Michael Laudrup succeeded
"I'm sad to be leaving, but it is in the best interests of both myself and the club," he said.
English Premier League
Manchester City boss