The 32-year-old England international, a £12million signing from Aston Villa in 2009, will enter the final year of his contract in the summer.
Barry has said he would like to extend his stay and even end his career at the Etihad Stadium but his contract situation has led to speculation about his future.
Manager Roberto Mancini is a big admirer of the player and, with fellow midfielder Yaya Toure tied to a new deal just last week, the Italian has suggested Barry could be next.
When asked about Barry at his latest press conference, Mancini said: "I think we have three or four players in this situation.
"We will decide this in the next month but these players are important players for us.
"I am very happy with Gareth because Gareth is a good guy and a serious professional player. He is really serious as a man."
- Sports & Recreation