According to the Mirror, Bale is keen to see where he fits into new Los Blancos manager Julen Lopetegui's plans, following the summer departure of former coach Zinedine Zidane and club legend Cristiano Ronaldo. The pair are set to meet this week and decide one way or another what lies in Bale's immediate future.
If the Welshman is unconvinced that he will be afforded a starring roll, then he could well look for an exit - with United ready to pounce.
However, Bale's exit could be dependent on Real Madrid's success in capturing their primary transfer target - Chelsea's Eden Hazard. Los Blancos chief Fiorentino Perez is believed to be unwilling to sanction Bale's exit without a suitable replacement, after already selling Ronaldo this summer.
It's believed though that Real could be willing to offload Bale as well as the Portuguese in order to make room for TWO new Galacticos. Hazard, as well as Kylian Mbappe, Neymar and Robert Lewandowski, are all on the list.
Hazard hinted on Saturday that he could look to leave Chelsea this summer, and intimated that everybody knows the identity of the club he wants to join, while Lewandowski has done similar multiple times in the past.
Former Spurs man, Bale, has endured mixed fortunes during his time at the Bernabeu, accumulating an impressive scoring record, but failing to win over the fans. During his side's domestic struggles last season, Bale was repeatedly targeted by the Los Blancos faithful, with boos and jeers greeting his every touch of the ball.
United are on the look out for a marquee signing this summer, and Bale could add a real injection of energy to the side as José Mourinho looks to challenge Manchester City for the Premier League title.
At the age of 28, Bale is arguably in the prime of his career, and could offer the Red Devils a potent attacking threat should they decide to invest.
Meanwhile, United defender Luke Shaw is rumoured to have turned down a move to Everton, as Mourinho desperately looks to offload the injury prone player this summer.
It is believed that the former Chelsea manager is eager to bring in Alex Sandro, and would use England international Ashley Young as backup, while Shaw would be sold to streamline the side.