According to reports in Spanish and British media, Real are poised to pay Manchester United 40 million euros (29 million pounds) to buy Spain keeper De Gea, who has not played this season because of the transfer saga, before the latest transfer window closes on Monday.
If the deal is confirmed, it will put Navas, who joined Real from Levante for a fee of around 10 million euros after a sparkling performance at last year's World Cup finals, in a difficult position.
Known as the "Falcon of Costa Rica" for his agility, the 28-year-old was one of the star players in Brazil and his brilliant form helped the central American nation unexpectedly reach the last eight.
Against Betis, he demonstrated that agility when he leaped to snatch the ball from the feet of forward Ruben Castro when he seemed certain to score and later saved a penalty from the same player with another fine stop.
Real's new coach Rafa Benitez has repeatedly declined to confirm or deny whether De Gea will be joining the club, where he is seen as a long-term replacement for former captain Iker Casillas, who left to join Porto last month.
Asked on Saturday whether Navas, who competes for a place with new signing Kiko Casilla, would definitely be staying, he told reporters: "What answer have I given the last three weeks to questions about the goalkeepers?"
"I am delighted with the three keepers and I want to enjoy the 5-0," added the 55-year-old, who took over from the sacked Carlo Ancelotti at the end of last season. "I work with the players I have. I am very pleased."
Navas earned effusive praise from Emilio Butragueno, a former Real striker who is now a club director.
"He was spectacular, he reacted like a cat," Butragueno told Spanish television.
"Ruben Castro totally had the advantage and it gives some idea of what kind of keeper he is," he added.
"Delighted and the fans were too. He participated a lot in the game, he influenced the result with his brilliance and you have to emphasise that.
"We don't talk about players from other clubs, we are delighted with the squad."
(Reporting by Iain Rogers, editing by Sudipto Ganguly)
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