The 25-year-old has been with Los Che since 2006, progressing through the youth system at Mestalla before making his first-team debut in 2012.
The Spain international has now featured on 258 occasions for Valencia in all competitions, including 31 outings during the 2020-21 campaign, providing one goal and five assists in the process.
Barcelona have long been linked with a move for the defender amid suggestions that the Catalan giants are keen to bring in competition for Jordi Alba at left-back.
However, according to Superdeporte, the former Spanish champions have made it clear that Gaya will not be available for transfer under any circumstances in the upcoming market.
The report claims that the club also want to sign the Spaniard to a new deal, but there is no immediate rush as he still has more than two years left to run on the existing agreement.
Juventus have also recently been credited with an interest in his services.