The film is Cooper's next directorial effort following 2018's A Star Is Born. Cooper will produce, co-write with Josh Singer and star as Bernstein.
Bernstein is behind musical hits such as West Side Story and Candide, among others. The biopic will span 30 years and focus on Bernstein's relationship with his wife Felicia Montealegre.
Cooper originally had the film set up at Paramount and has been working with Bernstein's children Jamie, Alexander and Nina on the project.
Netflix will release the film in theaters before it hits the company's streaming service. Production is expected to begin early next year.
"His strong passion and clear vision for bringing to life the relationship between the iconic composer Leonard Bernstein and his wife Felicia Montealegre promises to do their lives justice as only this renowned filmmaking team can. We are honored to be working with the Bernsteins to share their family's story with audiences everywhere," he continued.