Producers Julian Schlossberg and Letty Aronson announced the project Friday.
Featuring a book by musician and avid jazz fan Allen, the show will feature existing music from 1920s New York.
The show's creative team, run dates, theater and casting will be announced shortly, the producers said.
The story is about a young playwright newly arrived on Broadway and forced to cast a mobster's talentless girlfriend in his drama in order to get it produced.
Allen's 2011 comedy "Midnight in Paris," which is partially set in 1920s Paris, is up for Best Picture, Best Art Direction and Best Original Screenplay at Sunday's Oscars. Allen is also nominated for Best Director.