Tuttomercato reports that the Dutch side are looking to bolster their midfield this summer and are keen to sign Gago.
PSV sold Ibrahim Afellay to Barcelona in January and are keen to bring in a replacement for the midfield this summer.
The 25-year-old has fallen down the pecking order at Real Madrid and has long been linked with a move away form the club.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.