The 32-year-old has just over 12 months remaining on his £100,000-a-week contract at the Etihad Stadium, and following the arrival of Manuel Pellegrini, could be moved on during the transfer window.
Gunners boss Arsene Wenger is a known admirer of Barry, and tried to lure him to the Emirates Stadium when he left Aston Villa to sign for Manchester City in 2009.
It's thought that a bid of around £3.5million will be enough to tempt City to sell, though Arsenal are unlikely to match the England international's current wages, and hope that the security of a two-year contract offer will convince him to head to the capital.
A Sun insider said: "It looks like Arsenal are just weighing things up at the moment and they do have other priorities. But they know Gareth would do a good job for them."
Tottenham are also in the market for a holding midfielder, as Andre Villas-Boas looks for a replacement for Scott Parker at White Hart Lane, and could make a rival bid for Barry's signature.
However, as Spurs look set to complete the signing of Brazil international Paulinho from Corinthians, in a big-money £17.1million deal, their interest in the Manchester City star might now have ended.