Bale has been a long-term target for United since joining Madrid from Tottenham Hotspur in 2013, and poured doubt over his future at the club in the immediate aftermath of his match-winning cameo in the Champions League final earlier this year.
United boss Jose Mourinho and executive vice-chairman Ed Woodward still believe that the 28-year-old would be an ideal addition to their attacking lineup, and the Manchester Evening News reports that they have made progress in negotiations with Madrid.
Bale has grown frustrated with his lack of first-team football at the Bernabeu in recent seasons, although Zinedine Zidane's departure could still have an impact on the Welshman's future.
New boss Julen Lopetegui will need to convince him that he is a major part of his plans for the club in order to keep hold of him, although Bale's preference is still thought to be staying at Madrid.
However, with speculation linking Madrid to the likes of Neymar, Kylian Mbappe and Eden Hazard, Bale's status remains uncertain and it is thought that the Welshman will be allowed to have the final say on whether he stays at the club or not.