The Old Vic will also see James Earl Jones and Vanessa Redgrave reunite in Shakespeare's Much Ado About Nothing.
The pair, who last appeared together in Driving Miss Daisy, will play troublesome lovers Benedick and Beatrice in the play directed by Mark Rylance.
Cattrall will star in the story of a failing film star who quits Hollywood when her big comeback is a flop.
The Old Vic's spring programme also includes a version of Terence Rattigan's The Winslow Boy.
Old Vic artistic director Kevin Spacey said: "The Old Vic has always been first and foremost an actors' theatre, a home for great talent and memorable performances.
"We're so proud of where we have gotten to in nine seasons, and for me this new season of work shows just how far the Old Vic has come.
"What an exciting prospect to see the wonderful Mark Rylance direct supreme actors Vanessa Redgrave and James Earl Jones in Much Ado About Nothing.
"We are honoured that Mark is bringing his unique talent to the Old Vic.
"I am delighted to welcome Lindsay Posner back to our theatre with The Winslow Boy and I know that audiences will be thrilled at the prospect of Kim Cattrall taking on Sweet Bird Of Youth directed by Marianne Elliott, whom we are so excited to welcome for the first time."