The 32-year-old signed a five-year deal when he arrived from Aston Villa in 2009 and wants to extend his stay at the Etihad Stadium beyond the summer of 2014.
Quoted in several national newspapers, Barry said: "I am happy here and very settled.
"I spent 12 years at Villa and I've spent four years at City and I'm not really one for moving around. I'd like to stay as long as possible.
"For a player to feel settled and feel very comfortable, you generally want to be playing regular football - and I've been lucky enough to feature in a lot of games this season.
"That's one reason why I am happy to stay."
He added: "I am sure decisions will be made over the summer."