The theatre facilities include a bar, two rehearsal spaces, library and costume hire.
LDS produce around 200 productions a year, with each season consisting of 12 plays and 1 pantomime.
Acts have included Prunella Scales, William Roache, Topping and Butch and Charles Dance.
Leicester born playwright Joe Orton and actor Richard Attenborough are both former alumni from the society.
The expansion would result in 2 new rehearsal rooms, new studio, new workshop, new box office and foyer as well as providing better disabled access.