The Daily Star says France legend Zinedine Zidane, a sporting director at Real, tracked Hazard's development at Lille with interest.
The Belgian made a £32m switch to Stamford Bridge last summer but now Zidane wants him at the Bernabeu.
He has recommended Hazard to Real chiefs with a view to a move, according to sources in Spain.
Madrid are so keen to land Hazard they would reportedly be willing to let Angel Di Maria join Chelsea in a cash-plus-player swap deal.
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