According to La Nacion (h/t ESPN FC), talks with the Costa Rica international started before the 2018 FIFA World Cup, when new manager Julen Lopetegui "identified Navas as a cornerstone of his squad."
The 31-year-old has looked primed for the exit door for much of the summer transfer window, as Courtois has looked set to replace him.
Simon Jones and Jonathan Spencer of MailOnline recently reported Real had got as far as agreeing personal terms with the Chelsea stopper.
Los Blancos also signed 19-year-old Andriy Lunin in June, apparently to join Kiko Casilla as a back-up to the player brought in to replace Navas as No. 1.
However, if Real are looking to sign Navas up to a new deal, then Courtois may yet remain at Chelsea—there is no way the pair could play together in the same squad.
Per La Nacion (via ESPN FC), Navas is looking to ensure that he is tied to Real until he is 34.
His deal at the Santiago Bernabeu only lasts until June 2020, when he will be 33, so he is angling for at least a yearlong extension.
If and when Navas' contract is announced, it will effectively shut down any further talk of Real bringing in a new No. 1—they are also interested in Tottenham Hotspur's Hugo Lloris, per Jones and Spencer.
And it will mark something of a turnaround given that Real have long been thought to want a new, better goalkeeper:
It will also be just reward for a man who has played a key role in Real's three UEFA Champions Leagues triumphs in a row:
Navas succeeded the legendary Iker Casillas in the Real goal but was then almost sent to Manchester United as part of the deal to sign David De Gea back in 2015.
It looks now as though he could have survived another bout of reports calling time on his Real career and that he will continue on into the Lopetegui era.