The Sun says Real head coach Zinedine Zidane has cooled his interest in Monaco's teenage striker Kylian Mbappe following the rapid development of Marco Asensio and will focus on strengthening other areas of the team instead.
The 12-time Champions League winners have been persistently linked with De Gea and came close to signing the Spain international in 2015, but the transfer collapsed on deadline day as the necessary paperwork was not filed on time.
A £46m transfer would make De Gea the most expensive goalkeeper in football history ahead of Ederson Moraes, who moved to Manchester City from Benfica for £34.7m earlier this summer.
However, United are highly unlikely to sanction the sale of De Gea with less than two weeks remaining in the transfer window.
De Gea played the full 90 minutes of United's 4-0 win over West Ham on 13 August. He has two years left on his contract at Old Trafford.
Red Devils manager Jose Mourinho said last month that he was "100%" certain that De Gea would still be a United player when the transfer window closes.
"I can guarantee that he's not going this season, that I can, and my feeling is it will be very difficult for him to go. Because he's a very honest boy, very straight," he was quoted as saying by the Guardian.
"He was contacted for a long time [by Real]. The club was close, then we open because I always have this feeling of when a player has a desire to go I don't like to stop players to go because in the end you don't get what you expect from them if they want to move and they don't.
"I don't think the feeling from him [towards Real] is very good. I see him very happy and focused and working better than ever so for me 100% he stays with us."