Meryl Streep is also talking with the Disney production company about joining the film, the show business newspaper said. She would play the story's evil, ugly, spell-casting witch.
Disney officials and the actors' representatives had no immediate comment on the report.
The BroadwayWorld.com news website reported in October Streep, who has won three Academy Awards, was in talks to play the witch in the film.
Rob Marshall, who directed Depp in Disney's fourth "Pirates of the Caribbean" film, "Pirates of the Caribbean: On Stranger Tides," will direct "Into the Woods," Disney announced last year.
Marshall will also produce the film with John DeLuca through their production company, LUCAMAR Productions, Disney said.
The film would be the second Sondheim musical adaptation Depp has worked on. He was also in the horror musical "Sweeney Todd: The Demon Barber of Fleet Street," directed by Tim Burton, and won a Golden Globe Award for Best Actor.
It is expected Depp will sing in "Into the Woods," Variety said.
The musical centers on a childless baker and his wife. who try to lift a family curse by journeying into the woods to confront the witch who cast the spell on them.
Along the way, they meet up with "Little Red Riding Hood," "Jack and the Beanstalk," "Rapunzel" and "Cinderella" among other characters.
Depp will play the baker, Variety said.