Reports from Spanish newspaper would suggest that Manchester United have accepted a fee for their young star, which could see 28-year old Navas move in the opposite direction.
Navas who joined Real on a six-year contract last summer has postponed his trip to the United States, where he is due to join Costa Rica in training ahead of Saturday's international friendly against Brazil in New Jersey, until Tuesday.
De Gea is under contract with United for another season and has been left out of Louis van Gaal's squad -- with Sergio Romero installed as No. 1 -- since his last appearance on July 29 against Paris Saint-Germain.
He is set to arrive at Las Rozas on Tuesday evening to join the Spain team in training and Spain coach Vicente del Bosque is among those keen for the goalkeeper's future to be decided as soon as possible.
"He is active, training and ready to play," Del Bosque said. "I hope the situation is resolved from Sept. 1. If the situation drags on, then we could take another decision but at the end of August it's not significant."
De Gea was included in Spain's squad for Saturday's Euro 2016 qualifier against Slovakia in Oviedo and for the match against Macedonia in Skopje three days later.