The 19-year-old left back has enjoyed a standout year for Los Che, who want to renew his contract and increase his release clause, which currently stands at £13 million.
Spanish publication AS has said that Real have sped up the process in order to make him their number one target for the summer, having already completed the signing of Porto right back Danilo for a reported £23 million.
City are the most interested out of the Premier League clubs, with Arsenal also thought to be considering an attempt to make a deal. The Manchester side have identified him as a potential replacement for Aleksandar Kolarov, who is thought to be keen on a move back to Italy.
Gaya has made the left back spot his own at Valencia this season after making the step up from the second team last year, featuring once in La Liga and three times in the Europa League. He has missed just one league game this season through suspension, and scored in a 3-0 home win against Cordoba earlier in the campaign.
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The youngster suffered a hamstring injury in training with the Spanish under-21 side, but is expected to make a recovery to start against Villarreal at the Mestalla this weekend.
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