Throughout this summer, United were frequently linked with a move for the winger, who endured an injury-hit 2016-17 campaign for the La Liga giants.
No deal materialised, and Bale has continually stressed that he remains happy in Madrid, but according to The Sun, United are likely to maintain their interest in the Welsh star.
While Real are said to have rebuffed approaches during the most recent transfer window, officials of the Bernabeu club have allegedly informed their United counterparts that a significant bid would be considered in 2018.
Bale still has five years remaining on his contract, while the 28-year-old has started both of Real's opening two league fixtures with Cristiano Ronaldo currently suspended.
It has been suggested that Real intend to give a more active role to Marco Asensio, who is quickly emerging as one of Spanish football's top prospects.