Real and United had tied up the deal but blamed each other after it collapsed because the paperwork was not filed until shortly after the deadline.
Costa Rica international Navas remains at Real and United announced on Friday that Spain's De Gea had agreed a new four-year deal with the Premier League club.
Benitez told a news conference ahead of Saturday's La Liga match at Espanyol he had full confidence in Navas, as well as support keepers Kiko Casilla and Ruben Yanez.
He said he had been having dinner with Navas and the keeper's wife and agent the night the transfer failed to go through.
"He (Navas) knows how much confidence I have in him and he is calm and confident," Benitez told reporters.
"All I want is for him to do things really well," added the 55-year-old, who took over from Italian Carlo Ancelotti at the end of last season after Real failed to win any of the three main competitions.
"Yesterday he trained superbly. He made three or four great saves. Beyond that he is not affected. He is delighted and so am I. I always tell the truth."
Real, who play their Champions League Group A opener at home to Shakhtar Donetsk on Tuesday, will be without Colombia playmaker James Rodriguez and Brazil fullback Danilo for several weeks after the pair returned injured from international duty.
Benitez, who said he had enough quality in his squad to cope, is likely to deploy Spain midfielder Isco and his international team mate Dani Carvajal in their place.
Real drew their opening match of the campaign 0-0 at promoted Sporting Gijon before thrashing Real Betis, the second-division champions, 5-0.
They are two points behind joint leaders Barcelona, Atletico Madrid, Celta Vigo and Eibar, who all have six points.
(Reporting by Iain Rogers, editing by Ed Osmond)
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