De Gea, 27, almost joined the Spanish giants in 2015, however, a late fax ended any hope of a deal.
The Spanish news outlet say De Gea is keen to return to Spain and hopes to join Real this summer.
However, any deal is likely to hinge on United signing a replacement, with Jose Mourinho keen to sign Atletico Madrid stopper Jan Oblak before letting De Gea head to Real.
Oblak has been heavily linked with a move to the Premier League in recent months, with the Red Devils and Arsenal said to be tracking the Slovenia international, 25.
Diario Gol also say De Gea’s agent, Jorge Mendes, has held talks with Real president Florentino Perez over any potential move to the Spanish capital.
If De Gea impresses at the World Cup in Russia this summer, that could force Real’s hand and they could table a formal offer for him in the coming weeks.
Real Madrid full-back Marcelo believes it is “inevitable” that Neymar will join Los Blancos and has said “Cristiano Ronaldo doesn’t own Real Madrid” in reaction to the Portuguese’s presence seemingly posing a problem preventing his fellow Brazilian’s arrival.
Madrid want Ronaldo to stay at the Bernabeu but “the door is always open” for Neymar, Marcelo has said.
“Cristiano Ronaldo doesn’t own Real Madrid, if the president chooses – anyone can be there,” said Marcelo.
“There’s no such thing as Neymar not being there because Cristiano is there.”
Sky Sports’ Spanish football expert Guillem Balague said Real may be willing to offload Ronaldo and Gareth Bale in order to buy former Barcelona forward Neymar from Paris Saint-Germain.
Both Ronaldo and Bale hinted they could leave Real after the 3-1 win over Liverpool in the Champions League final last month.
But Marcelo hopes the Ronaldo will remain a teammate and that he will form a formidable attacking line with his compatriot.
“All of the players want Ronaldo to stay at Real Madrid, but one thing is not related to the other. The door is always open for Neymar to join.
“I think the best players in the world have to play at Real Madrid, and I already said for two or three years that, for me, one day Neymar will be there.”
The president of third-division Spanish side Real Murcia says he tried hiring Guti as manager but is now convinced the Spaniard will be appointed to that role at Real Madrid.
Victor Galvez claims he was speaking with Guti at a restaurant right as news broke of Zinedine Zidane‘s decision to resign as manager of the three-time reigning Champions League winners, bringing an end to their meeting.
“I spoke to Guti and I was going to see him on the Thursday, I wanted to bring him to Real Murcia but Zidane left Madrid…if only they had waited several days to announce it,” said Galvez.
“It’s down to Florentino (Real Madrid president Florentino Perez) to decide and announce the new coach and if it’s Guti then great,” Galvez said of Madrid legend Guti, who played for 15 seasons at the La Liga club.
“Guti will be the Real Madrid coach and so he won’t be coming to Murcia. I only wish Zidane had stayed a few days longer.”