The Spanish giants will pay 103m euros (£88.5m) for the 19-year-old, in addition to various potential add-ons.
If those add-ons are achieved, the deal could reach 133.9m euros (£115m).
Bellingham, who started his career at Birmingham City, will be presented as a Real Madrid player at a ceremony on Thursday.
“Thank you to everyone at BVB [Dortmund] and to the fans for everything over the past three years,” Bellingham said.
“It’s been an honour to wear your jersey so many times, in big and small moments.
“Even though I look forward to my next destination, I will never forget the journey there. Once a Borusse, always a Borusse. All the best for the future.”
One of England’s top performers at last year’s World Cup, Bellingham was named the Bundesliga’s player of the season as Dortmund missed out on a first league title in 11 years on the final day.
The initial fee means he becomes the second-most expensive English footballer after Jack Grealish, for whom Manchester City paid Aston Villa £100m in 2021.
Bellingham is also Real Madrid’s second-most expensive signing, after Eden Hazard’s 115m euro move from Chelsea in 2019, and the third-most expensive teenager in history.
Paris St-Germain paid Monaco 180m euros for France forward Mbappe in 2018, while it cost Atletico Madrid 126m euros to sign Joao Felix from Benfica in 2019.
Bellingham excelled at Dortmund where, in October last year, he made history by becoming the club’s youngest captain aged 19.
He played 42 times for his club in 2022-23 – scoring 14 goals and registering seven assists.
“We thank Jude for three years of passion for Borussia Dortmund. It was a fantastic time together,” Dortmund chief executive Hans-Joachim Watzke said.
“We would also like to thank Real Madrid for the always fair and constructive discussions.”
The £25m Dortmund paid Birmingham in 2020 made Bellingham the most expensive 17-year-old in the history of football.
Following that deal, the Championship side were mocked by some for retiring the teenager’s shirt number despite him only playing one full season of professional football.
Bellingham has been included in Gareth Southgate’s latest England squad for the upcoming Euro 2024 qualifiers against Malta and North Macedonia, but will not feature in the games.
The player will instead continue his rehabilitation in the England set-up as he recovers from the injury that kept him out of Dortmund’s final game of the season.
