Arsenal were interested in the player to add cover in the left-back position, with Kieran Gibbs blighted by a series of injuries and Nacho Monreal unable to live up to the standards held by the Gunners. Their defensive record this season has been less than exemplary, which has raised questions over the players, something that Arsene Wenger is willing to rectify.
The north London club have already signed Gabriel Paulista from Villareal and were exploring other options to shore up their defence, but are likely to be thwarted by Real Madrid, who are willing to pay any fee to secure his services.
Arsenal currently have Hector Bellerin, Calum Chambers, Per Mertesacker, Laurent Koscielny, Gabriel, Gibbs and Monreal to oversee their defence, with Mathieu Debuchy out with a long-term injury after dislocating his shoulder against Stoke City.
The Gunners are currently active in three competitions -- Premier League, the Champions League and the FA Cup -- and the FA Cup holders have progressed into the fifth round of the competition.
Wenger is always on the lookout for young talent who can add dynamism to his squad, as displayed by his signing of 17-year-old Krystian Bielik in January. However, with Gaya's fees spiralling out of control, the Gunners are unlikely to be interested, especially with the Frenchman having to address other pressing concerns in midfield in the future.
The defender's contract expires in 2016 and Valencia are doing their best to tie him down to a longer deal, which would raise the fee and give the La Liga club more bang for the buck.
The 19-year-old's talent has been well documented by clubs across Europe, who are also keeping a tab on the developments but are yet to join in the race.