The 27-year-old is a key target for Tony Pulis, who has been searching for a specialist left-back since joining the club.
Allan Nyom, Chris Brunt, James McClean and Jonny Evans have all filled the gap on the left side of defence in the past two seasons.
Gibbs is likely to leave the Emirates this summer, having most likely dropped to third-choice left-back behind Nacho Monreal and new arrival Saed Kolasinac.
The Gunners want around £10m for the defender, but the two clubs are not yet close to a deal.