The Spanish giants are interested in the young midfielder, who has established himself as one of the most-sought after players in Europe
The 20-year-old midfielder is widely regarded as one of Europe's most promising talents and played an integral roll in helping Lille win the Ligue 1 title last season.
His efforts have caught the eye of a number of clubs across the continent, with Arsenal, Liverpool and Paris Saint-Germain all credited with an interest in the Belgium international in the summer transfer window.
Real Madrid director of football Zinedine Zidane is a self-confessed admirer of Hazard's qualities and the retired midfielder has been reportedly given the green light from president Florentino Perez to initiate talks with the French outfit.
However, signing Hazard could prove to be a difficult affair, with Lille president Michel Seydoux stating last month the young midfielder was worth twice as much as Samir Nasri, who joined Manchester City from Arsenal for a fee believed to be in the region of €28 million.