His contract at the Premier League club was due to expire next year but the Blues drove a hard bargain for the forward, who will reportedly cost Madrid an initial €100 million ($113 million), with another €45 million in potential add-ons.
Hazard - who led Chelsea to Europa League glory this season - is the fourth new arrival at Zinedine Zidane's Madrid ahead of the 2019-20 campaign.
Real Madrid finished third in La Liga this past season, finishing behind both Barcelona and Atletico Madrid as the club struggled to truly replace Cristiano Ronaldo following the Portuguese star's move to Juventus.
Hazard left Chelsea after 352 appearances, during which he scored 110 goals and assisted a further 92 strikes.
According to the same story, Chelsea FC are in the market to sign a versatile winger to fill Hazard's void and Bailey fits the bill.
"Leaving Chelsea is the biggest and toughest decision in my career to date", said Hazard in a Facebook post.
But the Wales worldwide will not be forced out and has so far shown no interest in the handful of clubs that have shown interest.
Chelsea have reportedly identified Valencia's Goncalo Guedes as Eden Hazard's long-term replacement and have offered £62m for the Portuguese global.
"This is the real one, the other three [signings] I'm not quite sure but Eden Hazard is really that player that can make the difference in a team and I hope he would be able to play at the same level he has been in the Premier [League]". After reaching the semi-finals in 2014, Chelsea have not gone past the last 16 in the last five seasons.