Spanish outlet Diario Gol reported the Red Devils are stepping up their interest in the Wales international, with the attacker having recently returned to Los Blancos' squad after injury (h/t Jack Otway of the Daily Express). United have twice tried to sign Bale during his career, but it could be "third time lucky" as manager Jose Mourinho attempts to strengthen his options.
Per Diario Gol, Mourinho has told his former side to alert Bale to the chances of moving to Old Trafford, with Real considering new players to revamp their starting XI.
The Welshman has previously spoken favourably about the Special One and United.
Per Otway, Bale said in August:
"I think they're getting there. Obviously by bringing Mourinho in, it's kind of boosted the club.
"It did them good to win a few trophies last year and I think they're getting back to where they belong.
"They gave us a difficult game and I see no reason why they can't be a force in the Champions League this season.
"Jose's a great manager, he's done everything in football, he knows what he's doing and I'm sure he's building a great team for what he hopes to achieve there. I'm sure they will do fantastic this season."
Manchester City's performance level has warped the perceptions of United's season, but the Red Devils sailed through the group phase of the Champions League and sit second in the Premier League.
Bale will surely have taken note of this, and he could be the missing part of the jigsaw puzzle at the Theatre of Dreams.
United have been functional in the league, but they are missing a creative ingredient. Signing a superstar attacker would alleviate concerns and also produce faith that Mourinho wants to entertain while winning trophies.
In other Red Devils news, Inter Milan have refused to count out a bid for Daley Blind, with the defender out of favour in Manchester.
Samuel Luckhurst of the Manchester Evening News reported the Serie A side have an ongoing interest in the Dutchman, with coach Luciano Spalletti not ruling out bringing the player to the San Siro.
"Blind? I talk about my players, because the points we win in the next few matches will be done with them," Spalletti said. "I prefer to talk about my players because it is with them that we will look to be successful."
Blind was brought to United by compatriot Louis van Gaal in 2014, as the former manager attempted to change the club's fortunes and philosophy.
The 27-year-old featured heavily at centre-back as a partner for Chris Smalling in 2015/16, but he does not possess the physicality or speed to be a regular at the summit of the English game.
Blind would suit the pace and technical nature of Italian football, and Inter could be the perfect club for the versatile talent.