The England international has less than 12 months left on his City contract, with no sign that the club's hierarchy are prepared to offer the midfielder a new deal.
With fears over playing time under new boss Manuel Pellegrini, the 32-year-old is thought to be mulling over a move away from the Etihad during the close season.
Gunners boss Arsene Wenger is ready to pounce in an effort to add experience to his squad and values Barry at around £4m, according to the Daily Star.
Arsenal are reportedly ready to offer Barry a 12-month contract with wages of around £75,000 a week, with an option to extend the deal for a second season if successful.
Barry made 41 appearances for City last season, scoring twice.