However, eggheads of the Stamford Bridge are optimistic that neither the Los Blancos nor Camp Nou dwellers will meet their £200million valuation of their prize asset, who still has 24-months left on his present deal with Chelsea.
Courtois won the Golden Glove at the World Cup as Belgium finished third, impressing in Russian Federation with some top saves.
Chelsea are set to suffer a double blow less than four weeks before the start of the new season as Real Madrid close in on two of the club's star players.
With the Belgian leaving, the Blues are already looking around for replacements.
Some of the recent transfer speculation involving Chelsea...
Chelsea are ready to make a move for Petr Cech.
Former Chelsea player Craig Burley believes that Eden Hazard "needs" to sign for Real Madrid this summer.
Chelsea have also agreed a fee with CSKA Moscow for Aleksandr Golovin, says The Sun, with the midfielder rejecting an offer from Monaco to head to Stamford Bridge.
Courtois, who won the Golden Gloves for his performances at this summer's World Cup, has made no secret of his desire to return to Madrid at some point, having spent three years in the city with Atletico Madrid earlier in his career.
"After six wonderful years at Chelsea, it might be time to discover something different".
Hazard who scored three times in Russian Federation, added: 'I can decide if I want to stay or go, but Chelsea will make the final decision - if they want to let me go'.
Do you think Courtois would be a good signing for Real?
Giroud became a key player for the Blues last season but the Mirror claim Sarri is prepared to let him go.