The European champions are preparing for a new era under manager Xabi Alonso and they are expected to make several signings.
The Sun now claim that Los Blancos are eyeing a long-term successor to Thibaut Courtois and prefer to sign him this summer.
Raya, 29, is one of the top names on their wishlist, but it could be difficult to convince the Spaniard to accept a deputy role at Madrid.
Brighton & Hove Albion's Bart Verbruggen is also highly-regarded by Madrid, but the Dutchman may hold a similar stance.
Verbruggen has become a regular for club and country this campaign and may not want to reduce his minutes next term.
Espanyol's Joan Garcia is the other name on their radar, but the Spaniard could be on his way to Arsenal before next season.
Garcia agreed personal terms with the Gunners last summer. A deal could be straightforward when the transfer window reopens.