United are long-time suiters for Bale, who opted to move to Madrid instead of Old Trafford when he left Tottenham in September 2013.
The Spanish giants are ready to offload the 28-year-old after growing tired of the amount of time that he has spent on the treatment table in the last two seasons.
Real president Florentino Perez could also be planning to finance a move for Brazilian superstar Neymar, who has failed to settle at Paris Saint-Germain.
And it has given United some hope that they might be able to push through an unlikely deal for Bale before the winter transfer window closes at the end of this month.
If United boss Jose Mourinho does make a move, Bale’s former club Tottenham will have 72 hours to match it and discuss personal terms under the conditions of his sale.
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Bale has managed just six league appearances for Los Blancos this campaign but has registered two goals and one assist.
Meanwhile, Mourinho has been particularly impressed with one of his own players this season.
Jesse Lingard has flourished in his new central role and Mourinho reserved special praise for the 25-year-old.
“He’s in a moment of confidence where things go well for him,” he said.
“He’s a good professional and a fantastic kid in the dressing room.”