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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.