The Spanish champions have been searching for a new shot stopped ever since Iker Casillas left the club in 2015.
Real Madrid almost landed De Gea that summer after they agreed a deal with Man Utd.
However, a faulty fax machine made the move fall through at the final hour and Keylor Navas has kept goal since.
But his time in the Spanish capital looks set to be up at the end of the season as club president Florentino Perez looks to ring the changes.
And Real apparently want De Gea, or Chelsea’s Thibaut Courtois, to take over the reins, according to Sky Sports journalist Guillem Balague.
“I don’t think a new [Man Utd] contract is in the equation just yet,” Balague said when speaking about De Gea.
“It’s definite that Real Madrid have got De Gea or Thibaut Courtois as a target for next season.
“I think Courtois is easier. De Gea actually told team-mates in the national side that he was going to Real Madrid, as you’ve seen in the last couple of years, it’s not as easy as that.
“Courtois would also be happy to go to Madrid, and Real have it in their mind to go for one or the other.
“PSG are also after Courtois, so we’ll have to see what happens with that.”