The Blues started the campaign as Premier League champions but, for the second time in three years, mounted a lousy title defence.
Several players could depart with Alvaro Morata, Eden Hazard and Willian all being linked with moves away.
Yet Courtois, whose contract expires in 2019, could be the biggest of the lot.
Real are said to think that, if Liverpool see Chelsea miss out on a top four spot, the player will sign. The Blues need Liverpool to lose to Brighton and win their own match against Newcastle.
El Gol Digital say club president Florentino Perez has even made him his No 1 target for the summer window.
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Real Madrid may replace Keylor Navas with Chelsea star Thibaut Courtois
Ferdinand criticises Courtois' performance against Barcelona
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Meanwhile, back in December, La Liga expert Guillem Balague leaked news of Real’s Courtois interest.
“Real Madrid have told Thibaut Courtois' camp that they want to sign the Chelsea goalkeeper,” he said.
“The Spanish club have also heard that Courtois, whose two children live in Madrid, is willing to make a move to the Bernabeu and Real have decided to change their two goalkeepers before next season.
“Chelsea have prepared a Plan A and Plan B for the 25-year-old, and Plan B sees the Premier League champions ready to hand Courtois a new contract that would make him the highest-paid goalkeeper in the world.
“His current deal, around £100,000-a-week, expires in 18 months' time, and in May the Belgian rejected an opening offer that did not meet his double-my-money demands.”