Marco Silva has been active in the summer transfer market so far, having added Andre Gray, Richarlison, Tom Cleverley, Will Hughes, Nathaniel Chalobah, Kiko Femenia and Daniel Bachmann to his ranks.
However, the former Hull boss is still looking to strengthen his squad before the deadline and recent reports had suggested the Hornets were showing a keen interest in England international Gibbs.
The 27-year-old, who joined Arsenal in 2004 as a youth player, has been plagued by injuries over the past couple of years and he only managed 11 Premier League appearances last season.
First-team opportunities at the Emirates are expected to be further restricted for the left-back in this campaign and there have been several clubs, including West Bromwich Albion and Newcastle United, showing an interest.
Watford had joined the race for Gibbs' signature but they are now believed to have ended their pursuit after being put off by the hefty £17m price tag on the defender.