The 28-year-old is likely to leave the Bernabeu in the summer after an injury-ravaged 18 months and Madrid hope to recoup a large slice of the £87.5m they paid for him in 2013.
Bale chose Madrid over Manchester United when he left Tottenham but the Red Devils have consistently been linked with the Wales international.
Jose Mourinho admitted last summer that he was interested in signing Bale and was caught trying to persuade the winger to move to Old Trafford after a pre-season friendly.
However, Spanish publication Sport claim that Chelsea have entered the race to sign Bale and are willing to match United’s offer for forward.
Madrid are believed to value Bale at around £80m and Chelsea are said to be willing to pay a ‘good fee’ for the former Spurs man.
Bale is open to moving back to the Premier League after a trophy-laden five years in Spain.
More: Manchester United FC
Antonio Conte was far from pleased with his club’s transfer business last summer and is demanding significant funds to invest.
Chelsea’s title defence is all but over with the Blues currently 13 points behind leaders Manchester City having played a game more.
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