That is according to Spanish outlet Diario Gol, who say De Gea’s partner Edurne, who is a Spanish singer, is helping with the search for a new property.
The couple are looking in the area of La Moraleja which is thought to be in anticipation of a summer transfer.
Real have been searching for a permanent replacement for Iker Casillas who left the club in 2015.
Keylor Navas has been holding the fort in recent years but president Florentino Perez is said to be unconvinced in the Costa Rican.
So Real are preparing to launch a bid this summer to tempt De Gea from Old Trafford.
The 27-year-old came close to moving to the Spanish capital three years ago but the deal broke down because of a faulty fax machine.
But he would be expected to complete the move this time around when the La Liga champions come knocking.
But a move for the Blues keeper would seriously hinder any chance of a deal for Eden Hazard, who is also on their wanted list.