Sources tell Deadline Murphy will start shooting in December, ahead of a late 2020 release in cinemas and on the streaming service.
The project is one of several projects Murphy is developing for Netflix - these include the series "The Politician" and "Ratched", and a movie adaptation of another stage musical, "The Boys in the Band".
In "The Prom", Streep, who turned 70 on Saturday (June 22), will reportedly play a two-time Tony winner who teams up with Corden's character to rehabilitate their flagging careers by launching a campaign to support a high school senior banned from taking her girlfriend to the prom.
Key will play the principal of the Indiana school behind the ban, who falls in love with Streep's character.
Grande will develop the soundtrack alongside Murphy and her manager, Scooter Braun.