According to the Mirror, the 20-time champions of England are increasingly confident of finally landing the 28-year-old. However, Ed Woodward and Jose Mourinho fear Real will only agree to allow Bale to return to the Premier League should De Gea swap Manchester for the Spanish capital.
Bale joined Real over United in 2013 and has been relentlessly linked with a move to Old Trafford almost as a matter of routine every summer since then.
In the summer, Mourinho revealed he would "fight" to sign the Wales international if he were to suddenly become available this summer – during a time when Los Blancos were in contention to sign another high profile forward in Kylian Mbappe.
The United boss admitted defeat in that latest instalment of the transfer saga however, admitting any deal for the former Tottenham Hotspur star was dead following his side's 2-1 defeat in the Uefa Super Cup defeat to the Spanish giants.
In the tunnel ahead of the clash against Real Madrid, Mourinho told Bale: "I cannot buy you because you do not talk to me!"
However, while Mourinho seemingly conceded defeat, the rumours surrounding the Welshman's future have refused to go away, even though Bale signed a new five-year deal worth £350,000-a-week only 12 months ago.
Last month, Real Madrid president Florentino Perez explained United might have more luck should they return for Bale next summer, admitting "every player is transferrable right now."
"[Bale] is one of the best players we have and we are very pleased with him. There were no offers for him and what Mourinho said about him was just said in a complimentary way," Perez told Spanish radio station Cadena Ser.
"Every player is transferrable right now and we will see what happens in a year, although I don't know what'll happen next year, nobody does."
Similarly, De Gea's future has been subject of speculation for the last three summers, with Real Madrid determined to bring the Spaniard to the Bernabeu.
In the summer of 2015, the two clubs had reportedly reached an agreement that would have seen Real pay United £29m plus goalkeeper Keylor Navas for De Gea. The deal, however, collapsed after the paperwork was not submitted in time and the latter has since signed a new four-year deal with United.
The Mirror adds that while Mourinho is a keen admirer of Bale, who has won three Champions League since moving to Real, he would be very reluctant to part way with De Gea. The Spaniard has developed into a world class goalkeeper and while adding Bale would give United a genuine match winner, they would struggle to replace the Spaniard.