Courtois, 25, replaced Petr Cech as Chelsea's number one at the start of the 2014-15 campaign after spending three straight seasons with La Liga outfit Atletico Madrid between 2011 and 2014.
The Belgium international's current deal with the Blues will expire in the summer of 2019, and it is understood that initial talks over an extension have begun.
According to HLN, Courtois is aware that Real Madrid want to sign him, and the stopper will therefore demand a salary hike in order to become one of the best-paid goalkeepers in the world.
The report claims that the Belgian wants a £205,000-a-week contract to put him in line with Manchester United's David de Gea when it comes to earnings.
Courtois has kept eight clean sheets in 18 appearances for Chelsea this season.