The 19-year-old left-back, whose contract at the Mestalla runs until June 2018, is one of the brightest young talents in La Liga and has had an impressive season.
Spanish sports newspaper Marca reports on Wednesday that Real are looking to make a move at the end of the season for the Spain Under-21 international.
“We don’t have to send out messages to anyone,” Valencia’s director of football Francisco Rufete told Valencia TV on Wednesday.
“We are in talks since a few weeks ago with Gaya, wiith his family and with his agent to extend his contract.
“We are confident he will remain at Valencia for many years.”
According to the local media, Valencia want to tie Gaya to the club for a further six years and significantly increase his buy-out clause, which is currently set at ??13.4m.
Gaya, a product of Valencia’s youth academy, has been a regular in Nuno’s side since the Che club sold Spanish left-back Juan Bernat to Bayern Munich last summer.
The promising youngster has started in 17 league games, scoring one goal and setting up three more to help Valencia remain in contention for Champions League qualification next season.
Valencia striker Paco Alcacer, who on Tuesday extended his contract with the club until June 2020, is not surprised that Gaya is attracting interest from top clubs.
When asked about Real’s reported interest in Gaya, he told Valencia TV: “It’s normal that they are enquiring and that they want him.
“I see him very happy and well adapted to the team.”