Their interest in the Belgian represents the first proper contact ahead of a potential summer move.
The appointment of Hazard's idol Zinedine Zidane as Santiago Solari's replacement at the Bernabeu could prove decisive in making up the 28-year-old's mind.
Hazard has already described a move to Spain's capital as a "dream", with Madrid having frequently been linked with the former Lille winger over the last couple of years.
Chelsea are asking for around £100m from the reigning European champions, who are aiming to get the asking price down.
Madrid are keen to maintain a positive relationship with the west London club in the negotiations, as they did in the process of securing the signing of Thibaut Courtois from the Blues in 2018.
Blancos president Florentino Perez is hoping the reappointment of Zidane will lead to a summer rebuild, with Madrid also linked with Paris Saint-Germain duo Kylian Mbappe and Neymar.
Perez hopes that the presence of the France legend at the helm once more might tempt Mbappe to Madrid.
"Zidane’s French so maybe he could do something with Mbappe,” Perez, with a laugh, told reporters.
When pressed over whether he believed Mbappe could be convinced to join Madrid, Perez deflected, instead saying their focus was on finishing the season.
"Right now, all we are concerned with is ending the season well and start preparing for next season,” Perez said.
Neymar has also been rumoured as a Madrid target, with reports suggesting the club would shatter the transfer record PSG paid for the Brazilian while offering to make him the highest-paid footballer in the world.
It has also been reported that such a move would take Madrid out of the race, at least in the short term, for Mbappe, given the cost of landing both stars.
However, when asked who he would choose between the two PSG stars, Perez instead answered “Both.”
Hazard, meanwhile, is in a strong bargaining position over his future at Stamford Bridge as his Chelsea contact enters the final year.
The Blues have fought hard to keep their star man and had hoped to renew his contract - making him the club's highest-earning player in the process in an attempt to ensure he finished his career at Stamford Bridge.
With the Liga giants now stepping up their plans to sign Hazard, Chelsea are set for something of a turbulent summer.
As well as Hazard, there are also doubts over the long-term future of Callum Hudson-Odoi, who was targeted by Bayern Munich in January but was ultimately blocked from making the move to Germany.
He will also be entering the final year of his contract in summer 2019 and will likely be considering his future again as regular first-team opportunities in the Premier League under Maurizio Sarri continue to be hard to come by.
The Blues will also be hampered by the transfer ban that has been handed to them by FIFA, which is due to begin in the summer.
Football's world governing body rejected the Premier League club's attempt to delay the ban, that will span two transfer windows, with Chelsea now looking to appeal to the Court of Arbitration for Sport (CAS).