The 26-year-old was in impressive form for Los Che during the 2020-21 campaign, scoring once and providing seven assists in 33 La Liga appearances.
Barca have long been credited with an interest in the defender, as the Catalan giants are keen to bring in competition for Jordi Alba at left-back.
According to Mundo Deportivo, Gaya has put a stop to contract discussions at Mestalla, and Barca have been informed that Valencia could be willing to sell the defender for the right price in the upcoming market.
The Spain international has been with Los Che since 2006, progressing through the youth system before making his first-team debut in 2012.
Gaya, who is with his country preparing for this summer's European Championships, has two years left to run on his contract with the six-time Spanish champions.