It's not the first time that the Spanish giants have been linked with Hazard and former Real boss Zinedine Zidane wanted to sign the player before he resigned last summer.
Recently, Hazard hinted that he would be "torn" about his next move admitting that the lure of playing in La Liga could prove too much to resist. He is out of contract in 2020 so Chelsea may be tempted to cash in on their prize asset.
Onda Madrid reported that the 27-year-old could leave Stamford Bridge in a swap deal involving Croatian midfielder Mateo Kovacic, a deal which could be worth more than €170million. while Chelsea don't want to let Hazard leave, they are keen on a permanent Kovacic deal and the player is also happy in London.
Chelsea boss Maurizio Sarri this week told reporters that he was desperate to keep Hazard. "It's up to him. I think the club is available to sign a new contract immediately, but it's up to him if he wants to renew or not. I want Eden here, of course, but I want him here if he wants to stay here."
Those comments would seem to be at odds with Chelsea's "unofficial" policy of only handing one-year contracts to players over 30. Hazard will reach that milestone shortly after his current agreement expires and could prove problematic.