According to Don Balon, the La Liga titleholders will try again this summer, with club president Florentino Perez desperate to land the Spanish international.
The Red Devils stopper, meanwhile, is said to have told his agent Jorge Mendes to get the deal sorted.
Should Real sign De Gea, the Real president will look to offload first-choice keeper Keylor Navas to Paris Saint-Germain.
The latest situation is sure to increase the bad blood between United and Real, who are currently training at the same UCLA base in Los Angeles.
The rivals have been squabbling over who has the best facilities and hotels.