The actor has started making regular journeys to London for vocal coaching. Staff at Air Studios have confirmed Depp has attended the venue, but nobody has been allowed to hear him sing yet.
"Johnny has a closed set with just him and the director, but I've heard he can hold a note or two," a source said.
The big-screen version of Stephen Sondheim's creation will star Depp in the title role, and will also feature Helena Bonham Carter and Sacha Baron Cohen. Tim Burton has been signed to direct the film, which is due out next year.