The Spanish side are aware Jose Mourinho will not let his star man leave Old Trafford this summer.
And that’s because he is unhappy at Real for their handling of negotiations between the two clubs over Alvaro Morata, according to Don Balon.
United wanted to sign the striker but in the end opted to complete a £75m deal for Romelu Lukaku from Everton instead.
Florentino Perez is still desperate to sign another goalkeeper to at least provide competition for Keylor Navas though.
Real are now eyeing Athletic Bilbao starlet Kepa Arrizabalaga after a string of impressive performances last season.
The Champions League winners could sign the keeper for £26.5m (€30m).
De Gea came close to joining Real back in 2015 but for the arrival of a fax too late for the deal to be sanctioned.
But speculation has never gone away since then despite the 26-year-old signing a new contract at United after the incident.
De Gea is currently with the Red Devils in LA as they get their pre-season preparations underway.
Jose Mourinho will want the keeper to be at his brilliant best as he bids for Premier League and Champions League success.