Bale came off the bench to score twice as Madrid won the Champions League for the third successive season on Saturday, but after the game he admitted that he could leave the club this summer in search of more regular first-team football.
However, Madrid president Florentino Perez is known to be a keen admirer of Bale, and the Spanish giants have reportedly set a £200m asking price for the Wales international - equal to that which Paris Saint-Germain paid for Neymar last summer.
Madrid are hopeful that any sale of Bale - and possibly even Cristiano Ronaldo - would help to fund their own ambitious move for Neymar, who is thought to be unhappy in the French capital.
However, United would also need to match Bale's £650,000-per-week wages as he still has four years remaining on his contract, meaning that the price of signing the 28-year-old could rise to £320m.
The Express reports that United boss Jose Mourinho is concerned the club's owners will be put off by that figure, particularly as Bale will be 29 in July and will have little resale value at the end of his first contract at Old Trafford.
The likes of Chelsea and Paris Saint-Germain have also been linked with a move for Bale.