The Sun says Real will bid £22million for De Gea next summer.
Real boss Jose Mourinho and his 31-year-old skipper Iker Casillas have fallen out this season.
Spain Under-21 ace De Gea is now his top target.
A Real insider said: “De Gea is definitely wanted by Mourinho but a lot depends on his own future.
“But he’s studied De Gea’s form closely and is very interested.”