Elliott set up her new company, Elliott & Harper, with producer Chris Harper after leaving the National Theatre as associate director. Her production will open at the Gielgud Theatre in September 2018.
The piece will cast the lead character of Bobby as a woman – Bobbi – for the first time ever. Craig will play Bobbi while LuPone will play Joanne.
Company looks at what happens when Bobbi turns 35 and her friends and family wonder why she isn't settling down and hasn't found the right man. The show features songs including "The Ladies who Lunch", "Being Alive" and "Side by Side".
Patti LuPone added: ‘I saw War Horse in New York and The Curious Incident of the Dog in the Night-Time in London. I came out of both productions blown away by Ms Elliott's vision. So naturally I am thrilled beyond words that she wants me to be a part of Company.
"I thought I'd sworn off musicals, but working with Marianne was an opportunity I couldn't resist."
Company runs at the Gielgud Theatre from 28 September 2018.