The England international is thought to be surplus to requirements at the Etihad Stadium, and has been linked with a move away from Manchester with Tottenham Hotspur and former club Aston Villa supposedly interested in him.
However, the Gunners are currently monitoring Barry's situation at City, and he only has one year remaining on his £100,000-a-week contract, according to The Sun.
Arsenal wouldn't need to match Barry's current wages and could sign the player for as little as £3.5m.
"It looks like Arsenal are just weighing things up at the moment and they do have other priorities," an insider close to The Sun said. "But they know Gareth would do a good job for them."
Arsenal were previously interested in Barry five years ago before he joined City.