The Belgian stopper is currently mid-way through his second season on-loan at the Vicente Calderon from parent club Chelsea, and the La Liga side are reportedly desperate to keep hold of him for another year.
And the report claims they're prepared to make an offer to Chelsea to sign him permanently in the summer.
The European champions brought Courtois to Stamford Bridge from Belgian side Genk 18 months ago, and with Petr Cech the club's established number one, the young keeper was sent on loan to Atletico Madrid.
His performances in the Spanish capital have been superb, and the 20-year-old has become something of a cult hero among the fans at the Vicente Calderon - and the club don't want to lose him.
The report claims Chelsea now face a difficult decision - the west London club view the talented young keeper as the future successor to Cech as Chelsea number one, but Courtois doesn't want to return to Stamford Bridge to play second fiddle to the Czech Republic star next season.
Chelsea sent goalkeeping coach Christophe Lollichon to Madrid last weekend to check up on Courtois, and discuss plans for the future.
And Rafa Benitez is reportedly considering offering Courtois a new contract to ward off interest from Atletico Madrid.
He signed a new Chelsea deal in July 2011 - but is believed to want improved terms after establishing himself as one of the world's top young goalkeepers.