Since returning to the LaLiga club, Zidane has rotated his goalkeepers but Thibaut Courtois' injury has allowed Navas to stake his claim.
The Costa Rice international was in goal for Madrid's three consecutive UEFA Champions League titles won in Zidane's first spell.
But Madrid continue to be linked with a move for Manchester United goalkeeper David de Gea at the end of the season.
Navas had the gloves for Tuesday's (NZST) 1-1 draw at Leganes and he is planning to stay at the Santiago Bernabeu – if he is wanted.
"I'm very happy, grateful to God for giving me the chance to be here," Navas said after Madrid's poor form continued, Zidane's side having only won one of their past three LaLiga games.
"Whenever I get the opportunity to play I do the best I can. I hope I can stay at Madrid next season. I have a contract and we have to wait and see if anything changes there.
"It has been complicated season for everybody. We come out with a desire to win and play at our best but the results aren't always going our way, we're all sad for the situation, it's complicated.
"We're always willing to win and we've got our pride, our love for the shirt is worth a lot more, and we have to demand more wins and better results from ourselves."
"I think at this club every year we're under pressure, so it's no excuse. We have to be ready mentally to face any situation that happens and that's why we're Madrid players."