According to Marca's Jose Felix Diaz, they want Navas to stay at the Santiago Bernabeu for another year after his excellent recent form "completely renewed the trust" placed in him.
They want him to at least "fight for the possibility of being" the first-choice stopper even if De Gea were to come in this summer.
Diaz added that Zidane has repeatedly given Navas his backing, while the Costa Rican is also close friends with captain Sergio Ramos, who was "the first to hug" him after they retained the UEFA Champions League on Saturday.
After making an early return from Achilles surgery, Navas struggled somewhat for form earlier in the season, with ESPN FC's Dermot Corrigan highlighting one of his mistakes when Real played Napoli in February:
SB Nation's Lucas Navarrete took issue with his uninspiring clean-sheet record ahead of the Champions League final with Juventus, although Navas was not the only one to blame:
However, he was among Los Blancos' key performers in their run-in for the double, as he put in several impressive showings, including against Atletico Madrid in the Champions League semi-final and Malaga to seal La Liga on the final day.
Corrigan commented on his return to form:
He largely fared well against the Bianconeri on Saturday too. While he was slightly caught out by Mario Mandzukic's overhead kick, such was the quality of the strike it would be unfair to put much blame on Navas.
There's no doubting De Gea would be an upgrade, but Navas would make for a strong understudy.
Equally, if Real don't recruit a new goalkeeper this summer, the 30-year-old has shown that with confidence, he can be a strong stopper for his side.
With the backing of his manager and team-mates, Navas has an excellent platform from which to build and maintain that confidence, and with it his form should make him a viable option.