By Greg Stobart Follow on Twitter
The Spanish giants have made De Gea their top priority for the summer transfer window and hope to sign the goalkeeper as the long-term successor to Iker Casillas.
De Gea has been warned by Real Madrid that they are determined to sign a new first-choice goalkeeper and will move on to other targets if they do not sign him this year.
It is a similar tactic used by the club when they convinced Gareth Bale to force through his world record €100 million move from Tottenham in 2013.
De Gea, 24, has just one year remaining on his contract at United and has so far turned down the club's attempts to sign him up to a new deal on better terms than what he could expect to earn at the Santiago Bernabeu.
Goal reported on Wednesday that De Gea's representatives will hold fresh talks with United this week over a new contract, but the Spain international is under increasing pressure to engineer a summer transfer.
A move to Madrid would also boost De Gea's chances of replacing Casillas in the Spain national team ahead of the European Championship in France next season.
Casillas has two years remaining on his Real Madrid contract but has been booed by fans in recent seasons and is expected to either drop to the bench or leave the club this summer.
De Gea is a long-term target for Madrid and United manager Louis van Gaal expressed the club's fear that he will leave last week, saying "we shall already know who shall be the next goalkeeper. We also have a list of goalkeepers who can replace him because we have to be organised always."
De Gea was named United's Players' Player of the Season and Fans' Player of the Season at the club's awards ceremony on Tuesday.
Van Gaal, though, suggested after United's draw with Arsenal on Sunday that the former Atletico Madrid man was edging towards a return to the Spanish capital, particularly as his popstar girlfriend lives in Spain.
"A transfer is a process involving a player, a club and his present club," said Van Gaal on Sunday. "He has to choose. It is not easy to choose between two great clubs. We would be very happy if he stays. Of course he is Spanish and his girlfriend is Spanish and he can go to another great club, so it is difficult for him to decide.
"I don't think I can persuade him because he knows everything that he has at Manchester United at this moment. I don't have to say anything for the benefit of Manchester United because he can feel everything. The crowd has been unbelievable. You have seen that reaction here. It has been unbelievable, for the team and for him individually. When he is going he is losing that because here he has a lot of credits."