The Belgium international, into his seventh year at Stamford Bridge, has made no secret of his desire to play for the reigning European champions.
Hazard is about to enter the final year of his contract and, despite mixed messages in recent weeks, he does not look likely to agree fresh terms.
According to The Telegraph, both the 28-year-old and Chelsea are keen to get the issue resolved by the end of this month, though the ball is firmly in Madrid’s court.
Should Hazard get the assurances he wants from Real, he will reportedly tell his current employers that he wants to leave in the summer.
However, Chelsea want a nine-figure fee for their prized asset, even if it means running the risk of losing him for free next year.