The Spanish champions are ready to offload the Welshman at the end of the season due to the time has spent on the treatment table.
Despite missing 40 of Real’s last 60 games through injury, United will not be put off by the amount of time he spends on the treatment table.
Bale is poised to leave Madrid at the end of this season after five years at the Bernabeu, with United hoping to finally land the 28-year-old.
United missed out on Bale when he joined Real from Tottenham for a then world record fee of £85million in 2013, then again in 2016 before he signed a contract extension until 2022.
Spurs chairman Daniel Levy insisted on a first option buy-back clause that expires in summer 2019, but the club’s wage structure is lower than United’s.
Bale is currently out with a calf problem that is expected to sideline him for the remainder of the year.
Real have grown tired of Bale’s ongoing injury issues and are believed to be ready to offload the Wales star next summer.
United are convinced Jose Mourinho’s approach and handling of Bale could help the attacker develop more consistency in terms of his fitness.
Apart from his debut season, Bale was rarely injured during six years at Tottenham and there is a belief at United that he could get back to that level of regular football under Mourinho.
Mourinho revealed his desire to sign Bale on the eve of the season when he said he would be “fighting with the other coaches who would like to have him on their team” if he became available.
Despite Real’s willingness to listen to offers for Bale, United are likely to have to come close to matching the £85m they paid for him, given he has four years left on his contract.