Diario Gol says Courtois is an end of season target for Real to replace Keylor Navas.
Courtois has an agreement in place with the club to move at the end of the season.
Real are also expected to sign Kepa Arrizabalaga from Athletic Bilbao as they completely overhaul their goalkeeper ranks.
When asked if Courtois would like to move back to Spain to be with his family, Blues boss Antonio Conte made it clear he wanted the Belgian to stay at Stamford Bridge.
He said: “I don't know about this issue.
“When you are in a great club and you trust in the project, you want to stay. I think the family is not a problem."