The Spain international is a long-term target for the La Liga champions.
De Gea nearly joined Real in 2015 but a late fax ended any hope of his dream return to Spain.
And Spanish transfer outlet Don Balon say Real will make another bid for De Gea, 26, next summer.
They claim Real club president Florentino Perez has his heart set on landing the United star at all costs.
Don Balon add that Keylor Navas could be sold to PSG to make room for De Gea.
De Gea has been vocal about his Red Devils future, with the stopper claiming this week it’s a dream to play for United.
"Of course, you feel really proud when people think this about you," De Gea told MUTV.
"It’s really good, but I like to keep my focus, keep working hard and doing my best.
"To be fair, when you are really young, you don’t think about the future too much, you just want to play with your friends.
"When you get older, you start to dream about being there, about being at a top team.
"So of course it’s a dream to be at a team like Manchester United."
