The Red Devils have been regularly linked with a move to sign the Wales international over the past two or three seasons but a deal has failed to materialise thus far.
Bale struggled to hold down a regular starting spot in the Real Madrid team throughout the 2017-18 season after falling out of favour under manager Zinedine Zidane.
The 29-year-old came off the bench to score two goals in Real Madrid’s 3-1 win against Premier League side Liverpool FC in the Champions League final in Kiev last month.
Zidane stood down following Real Madrid’s 13th European Cup triumph to give Bale fresh hope of resurrecting his stalling career at the La Liga giants.
Sunday Times reporter Castles has claimed that Manchester United chief executive Woodward would like bring a player of the calibre of Bale to Old Trafford this summer.
“Bale is the kind of level of player Woodward wants to bring to the club,” Castles told the Transfer Window podcast.
“He is a hugely marketable player. He is confident of getting the Glazers to approve the huge finances involved in getting him.”
Manchester United made Brazil international Fred their first summer signing after the Red Devils agreed a £52m deal from Ukrainian Premier League champions Shakhtar Donetsk.
The Red Devils brought in Portuguese talent Diogo Dalot in a £17m deal from FC Porto earlier this week to strengthen their options at full-back ahead of next season.
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