This is according to The Sun, who claim Real Madrid have already made contact with the Blues regarding their world-class Belgium international.
Real Madrid are determined to make Hazard, 24, their star summer signing and are even prepared to send Bale in the opposite direction in order to secure the attacking winger’s services.
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Chelsea are under no pressure to sell their prized asset - he signed a new five-and-a-half-year deal worth £200,000-a-week in February - but might be tempted by the offer to sign Bale in exchange.
The 25-year-old, a two-time winner of the coveted PFA Player of the Year award, has endured a difficult second season at the Bernabeu. Many of the club’s fans have been critical of his performances this season and much of his fine work from last term appears to have been forgotten.
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Gareth Bale is a world-class performer
The Welshman, who became the most expensive player in football history when Real Madrid signed him for £86 million from Tottenham in the summer of 2013, scored 22 goals in 44 appearances during his first season with the Spanish giants.
More importantly, he also played an instrumental role in helping Los Blancos win both the Copa del Rey and their long-awaited tenth European Cup, while earlier this season he would also help the club win both the UEFA Super Cup and the FIFA Club World Cup.
Bale reportedly has little desire to leave Real Madrid and hopes to cement his status as the best player in the world at the Bernabeu.
But the opportunity to return to the Premier League, particularly if he discovers Real Madrid are prepared to offload him, could suddenly become an attractive option.
Real Madrid can sign David de Gea without Bale offer
Manchester United also have a strong interest in Bale, but lack a superstar performer that Real Madrid would want in return.
Goalkeeper David de Gea is certainly on their radar, but the Spain international is obtainable without the need to offload one of their Galacticos because of his contractual situation.
Chelsea unlikely to agree to deal
This leaves Chelsea, if Jose Mourinho opts to give the green light, in pole position to sign Bale.
However, the revered Portuguese coach has stated that Hazard would either cost £100 million per leg or “£100 million and one of Real Madrid’s best three players”.
Because of the way his career is progressing and his potential to become the best footballer on the planet, the chances of Chelsea agreeing to offload Hazard appear extremely slim, even if offered the chance to sign Bale in exchange.
Chelsea fans: would you swap Eden Hazard for Gareth Bale? Have your say in the comments below!