The Belgian playmaker has long been linked with a switch to Los Blancos but recently insisted that he is happy at Stamford Bridge, where his current contract runs until 2020.
Former Tottenham Hotspur striker Bale, meanwhile, is expected to depart the Bernabeu this summer and has reportedly asked to leave the club, with Manchester United thought to be monitoring his situation.
According to Don Balon, Madrid boss Zinedine Zidane is "desperate" to land Hazard this summer after a disappointing season that has seen the side fall 15 points behind Barcelona in the La Liga title race.
The newspaper says that should Zidane keep his job this summer, he will make acquiring Hazard his top priority and will sanction the exit of Bale in order to make the deal happen.
Previous reports have suggested that it would take an offer of Bale plus £100m in cash in order to persuade Chelsea to let Hazard go.