Gillan, will appear with Douglas Hodge in the show, which opens at the Donmar in October. Hodge plays a self-destructive lawyer, while Gillan will play his secretary.
The 23-year-old Scottish star will be seen later this year playing model Jean Shrimpton in her first major film for BBC Four We'll Take Manhattan.
In the original production of Inadmissible Evidence, Dame Eileen Atkins played Gillan's role in the original production at the Royal Court in London.
According to the Daily Mail, Jamie Lloyd will direct the Donmar's revival, which will make "subtle cuts" to Osborne's text.
Another British actress, Thandie Newton, is set to perform this autumn in a revival of Ariel Dorfman's play Death and the Maiden which will open at the Comedy Theatre in October. The play will see the 38 year old actress play a former political prisoner who kidnaps a man she believes raped and tortured her.
Newton famously appeared with Tom Cruise in Mission: Impossible II and won a Bafta back in 2006 for her work in Crash.
Other Doctor Who stars David Tennant and Catherine Tate are performing together in Much Ado About Nothing which has been the hottest ticket in the West End this summer.