The Blues are unbeaten in their last seven Premier League games following a run of six victories and one stalemate since a 1-0 win against Manchester United at the end of October.
According to Spanish outlet Diario Gol, Zidane has been granted permission by his president to make Chelsea an offer with the French coach regarding the Belgium international "as key to his rebuilding project at Real".
Hazard has previously spoken of his desire to be crowned the world's best player, and knows his chances of winning it will improve tenfold in the Spanish capital.
The report also notes that Hazard has in the past said it would be a 'dream' to play under Zizou.
When asked about contract for Hazard and Thibaut Courtois, the Italian boss said: "It is normal during the season to have the contact between the player - or the agent of the player - and the club".
Chelsea transfer news: Real Madrid ready to make offer for Eden Hazard
It's alleged Zidane contacted Hazard in the past week and comes after reports from England that Hazard has delayed signing a new deal at Stamford Bridge worth £300,000 per week.
"If you ask me about these two players, I will be very happy when they can have this deal but this is the responsibility of the club to try to do this".
Both Essien and Hazard played together between 2012 when the 26-year-old arrived from LOSC Lille until January of 2014 when Essien left the Blues for AC Milan as a free transfer. I think that Chelsea need to renew his contract.
But with the European champions struggling for consistency this season Zidane has decided he needs to freshen up his squad. He won't refuse to play for a full two years and then frankly who would want a player that out of shape and match fitness anyways.