Blues boss Antonio Conte wants Courtois to follow the example of Italian World Cup winner Gianluigi Buffon, who has spent 15 years at Juventus and achieved legendary status.
Real want Courtois after being forced to end their pursuit of Manchester United keeper David De Gea, and reports in Spain suggest the Belgian is interested in a move to Madrid.
But Chelsea are likely to offer him a new contract to convince him to stay – and Conte has challenged him to become an all-time great.
He said: “I think that a goalkeeper like Thibaut deserves to stay at Chelsea for many years and become a type of player to stay and write the story of this club, to become a legend at this club.
“I hope for Thibaut to repeat the same career as Gigi Buffon”
Chelsea manager Antonio Conte
“I hope for Thibaut to repeat the same career as Gigi Buffon. I think that Buffon, for me, is one of the best goalkeepers of all time.
“I stayed with him at Juventus before as team-mates, and then as his coach, in the national team also. I think Buffon is a legend for football, and I hope Thibaut has the same career as Gigi.
“Thibaut has the attributes to do it. He has great potential to continue to do great things, to reach great targets.”
Conte made 2006 World Cup winner Buffon his captain for both club and country and the 38-year-old has won seven Serie A titles with Juventus since they signed him from Parma in 2001.
Courtois, 24, has barely started his Chelsea career after spending three years on loan with Atletico Madrid following his £8m move to Stamford Bridge from Genk in 2011.
But despite him being constantly linked with a return to Spain, Conte insists he has no concerns about the future of a player who still has two-and-a-half years left on his contract.
The Italian added: “Thibaut is one of the best – if not the best – goalkeepers in the world. He stays with us and this is the most important thing.
“The other things are only speculation.
Real have previously sent Courtois a cheeky “get well soon” fax when he injured his knee in an effort to get in his good books.
But Conte added: “I think that was very good of Real Madrid, to send a message. Maybe, in the future, I’d like to do the same – with Ronaldo.”
Chelsea’s £33m misfit striker Michy Batshuayi is set to be on the bench against Stoke again today. But Conte revealed he has NOT asked to leave the club.
The Italian said: “He wants to stay, and I am happy for this. I think it’s very important for him as a footballer to complete this season here.”