"Courtois, decision imminent", Madrid sports daily Marca ran as its headline, claiming that the record 13-time European champions are "confident of his signing".
"The Courtois soap opera is coming to its end. The target, who yesterday did not attend Chelsea training despite being at the end of his holidays, speeds up the process to sign for Real Madrid, an announcement that could become official in the coming hours," Marca wrote.
Real will pay 35 million euros to bring the Belgium international, voted the best goalkeeper at the World Cup, to the Bernabeu, according to Marca.
"Courtois doesn't come to Chelsea training", noted Catalan daily Mundo Deportivo, for whom "his destination will be Real Madrid".
AS, another Madrid-based sports daily, said "Chelsea give Courtois up for lost and want (Athletic Bilbao's) Kepa Arrizabalaga to be his replacement".
It added the Premier League club "are willing to pay the 80 million euro release clause to Athletic" for the 23-year-old Spain international.
There is also speculation that Chelsea could try to sign Atletico Madrid's Jan Oblak.
Courtois, who spent three seasons at Atletico from 2011 to 2014, has repeatedly stated his desire to move closer to his two children in the Spanish capital.