According to Matt Law of the Daily Telegraph, the La Liga giants are on the hunt for a new goalkeeper this summer and have isolated Courtois as a possible target. As a result, the Blues are keen on extending the stopper's deal beyond its current 2019 expiry date.
In the piece it's noted that Madrid are also admirers of Manchester United man David De Gea, whom they came close to signing in 2015. The Red Devils star is said to earn around £200,000-a-week, something that may influence the Blues' negotiations with their man.
"The Belgium goalkeeper currently earns £100,000-a-week at Stamford Bridge, which is around half of what De Gea is paid by United and the Spaniard is likely to be given a rise if he joins Real," Law said. "Chelsea will, therefore, need to pay Courtois well over £150,000-a-week to tie him down to a new deal, and the 25-year-old may want to be sure of Real's intentions before committing himself."
Still, according to Kristof Terreur of Het Laatste Nieuws, the Blues have yet to officially offer the goalkeeper a new deal.
During the 2015-16 term Courtois, like a lot of Chelsea players, suffered a drastic loss of form, and there were some concerns about him going on to reach his potential. But under Antonio Conte, he's been back to his best again.
The rangy goalkeeper has helped the Blues build an impressive defensive record this season. Courtois has been a commanding presence at the base of the team, plucking crosses out of the air, excelling in one-on-one situations and distributing with calmness.
His shot-stopping is outstanding, too, as Courtois is so often on hand to paw out potentially goal-bound shots. Per WhoScored.com, his clean-sheet record speaks for itself:
It's imperative Chelsea keep Courtois on board. Although there are still some inconsistencies to his game, often Blues supporters take for granted the fact they have one of the world's best goalkeepers to call upon. If he was to move on at the end of the campaign, he'd be a difficult man to replace.
Real Madrid are a tempting proposition, though, and an undeniable step up, even from a club like Chelsea. Should Los Blancos decide Courtois is the man they want in goal next season, there'd be a temptation on the 'keeper's part to hear them out.
Still, there shouldn't be a rush to leave Stamford Bridge. After a season of disappointment, this term has seen the Blues reestablish their place at the summit of English football, and Conte will be desperate for them to make an impact in the UEFA Champions League next season. Losing their star goalkeeper would be a setback for those aspirations.