The 32-year-old England international has entered the final 12 months of his contract at the Etihad Stadium, and following the arrival of new manager Manuel Pellegrini, is concerned about his first-team future.
With the addition of Jesus Navas and Fernandinho already this summer, competition for places in the Manchester City midfield has already increased, and Barry might look to protect his long-term future with one eye on the 2014 World Cup in Brazil.
Gunners boss Arsene Wenger is willing to offer Barry a one-year contract with wages of £75,000-a-week, after offering him assurances that he will be an integral part of the senior team at the Emirates Stadium.
An option for a second-year extension will depend on his success at Arsenal, but it's anticipated that the player will accept a wage cut from his current £100,000-a-week salary.
Tottenham Hotspur and West Ham are both also believed to be monitoring Barry's situation this pre-season, but after opening negotiations with the Manchester City star, Arsenal remain favourites to secure his signature.