The 32-year-old signed a five-year deal when he moved from Aston Villa in 2009 and wants to extend his contract at the Etihad Stadium beyond summer 2014.
"I am happy here and very settled," the England international is quoted as saying in several national newspapers.
"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."
Barry has scored three goals in 53 international appearances for England.