It is understood that Gibbs is keen to play under Marco Silva at Vicarage Road.
West Brom had a £10m bid rejected for the 27-year-old and have since pulled out of talks, while Newcastle have also expressed an interest in the defender.
Gibbs made just eight Premier League starts last season after losing his place to Nacho Monreal and is yet to feature this season after missing the pre-season tour in the Far East.
The product of the Arsenal academy has 10 England caps to his name, and made his debut for the Gunners in 2007 after a short loan-spell at Norwich City.