Jonathan Kent's production will reportedly open in March at the Savoy Theatre, with the deal expected to be finalised "today or early next week".
Staunton plays Mama Rose in the show, which has music and lyrics by Jule Styne and Stephen Sondheim, and a book by Arthur Laurents.
The transfer would mark the first time Gypsy has been seen in the West End for forty years, when Angela Lansbury starred.
It would replace Dirty Rotten Scoundrels, currently playing at the Savoy Theatre, which announced a UK tour earlier this week.
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