The actors will swap roles throughout the play's run, which is slated to begin in February, the BBC reported.
Nick Dear is adapting Mary Shelley's 1818 horror story "Frankenstein; or, The Modern Prometheus" for the stage.
"Slumdog Millionaire" filmmaker Boyle previously directed Miller in the 1996 film "Trainspotting."
Cumberbatch is known for his work on TV's "Sherlock" and "Fortysomething."
Swim Week Miami Beach 2014 [PHOTOS]