The Borat creator is staring in the Tim Burton production of the musical alongside Johnny Depp and Helena Bonham Carter. He will play Signor Adolfo Pirelli, rival hairdresser to Depp's Todd, while Bonham Carter is the baker, Mrs Lovett.
Cohen had been asked to sing a number of songs but, after he struggled in rehearsals, Burton has decided to allow him to do them as raps.
“Pirelli is a notoriously demanding role with some difficult songs," a source told The Sun. “Any professional singer would struggle. So Sacha has been told to go for a rap style.
“There was never any question of Sacha being axed from the movie. He is going to do a fantastic job. But he couldn’t cut it with the singing. His voice was too low.”
The epic movie, a version of Stephen Sondheim's Sweeney Todd, The Demon Barber Of Fleet Street, is due for release next year.