Bale has been tipped to leave the European champions this summer amid reports that his relationship with manager Zinedine Zidane has worsened.
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According to the Express, United remain the favorites to sign the Wales international but they face competition from Premier League rivals Chelsea, including Bayern Munich and Paris Saint-Germain.
Real are said to have announced a £80m price tag on Bale, who came off the bench to score twice in Los Blancos’ 3-1 victory over Liverpool in the Champions League final.
Bale’s agent Jonathan Barnett is expected to fly to Madrid to meet with club president Florentine Perez to discuss the winger’s future this week.