Real have been linked with a move for the Belgian all season, and are said to have all but finalised a €100million (£86m) deal for the forward.
Hazard has fuelled speculation suggesting he would link up with Real Madrid manager Zinedine Zidane at the Bernabeu all season by refusing to commit his future to Chelsea.
Now L’Equipe report that an agreement for the 28-year-old was done weeks ago, but will only be made public once the Blues’ season officially ends.
That will come in Azerbaijan on May 29, when Chelsea face Arsenal in an all-London encounter and one of two all-Premier League European finals this season.
Hazard has been the brightest spark in an uneven season for Chelsea, where fans have often expressed displeasure at manager Maurizio Sarri’s selections and football style.
Despite this, Chelsea finished third in the Premier League to qualify for the Champions League, and have reached two major cup finals.
Hazard has only a year left on his Stamford Bridge contract and has not shown any inclination to sign a new deal with the club.
As a result, Chelsea are set to cash in on the Belgian despite his importance to the team and their imminent two-window transfer ban for breaking rules over the signing of under-18 players.