Madrid have been linked with a new goalkeeper throughout Navas's tenure at the club, but former manager Zinedine Zidane was understood to be a staunch supporter of the Costa Rican and saw no reason to replace him between the sticks.
However, Zidane's departure has opened the door for club president Florentino Perez to bring in a replacement shot-stopper, which is expected to be Chelsea's Courtois.
The Belgian is thought to be nearing a £31m move to the Spanish capital having entered the final year of his contract at Stamford Bridge.
AS reports that Navas accepts a new goalkeeper will arrive this summer, but that he has no desire to leave the club and will stay to fight for his place until at least the end of his contract in 2020.
The 31-year-old has been at Madrid since 2014, helping Los Blancos to three successive Champions League titles in that time.