The 19-year-old striker, born in Malaga to Scottish parents, has been at the Bernabeu since he was 13 and was once compared to Robin van Persie by former Scotland performance director Mark Wotte.
But Harper has opted to leave the Spanish giants and, having completed a medical with Brighton, agree a two-year contract, subject to international clearance.
Albion boss Chris Hughton said: “Even at the age of 19, Jack is a well-known name and having spent the past six years at Real Madrid, there is no doubting his pedigree.
“He has a lot of potential, is a skilful type, with a good eye for goal, and can operate as a striker or playing just off of a striker. We hope we can give him the platform to realise his undoubted potential.
“He has had an excellent grounding, having come through the youth system at Real Madrid – and also has international experience at youth level, having featured for Scotland’s Under-17 and Under-19 sides.”
Harper is currently recovering from a knee injury he sustained last season, and is at least eight weeks away from competing for a first-team spot.
Hughton added: “Jack’s next step is to move up to senior football, but his first priority is to complete his rehabilitation and work to get his match fitness; then challenge to breakthrough into our first-team squad.”