The Spanish giants are understood to have struck a deal that could rise to a club-record transfer once all bonus payments are taken into account.
After months of speculation, the 28-year-old Belgian striker will be officially unveiled on June 13, once he has undergone his medical.
Shortly after the transfer was confirmed, he posted a 700-word goodbye to Chelsea supporters on his Facebook page, explaining that it was his dream to play for Madrid since he was a young boy and leaving Chelsea was "the biggest and toughest decision in my career to date".
The fee is reported to be in excess of £130 million (AED 600 million) which will make him the most expensive transfer ever involving an English club, overtaking the £106.4 million Barcelona paid Liverpool for Philippe Coutinho in 2018.

New Zealand clinch ODI series against West Indies
Federer to be inducted into International Tennis Hall of Fame
Tommy Fleetwood set for 2026 Hero Dubai Desert Classic
Iraq defeats UAE in World Cup playoff decider
Bergs sends Belgium past France into Davis Cup semi-finals
