Stevens, who plays aristocrat Matthew Crawley in the ITV period drama, will appear in The Heiress from October.
It is an adaptation of the Henry James novel Washington Square, which became an Oscar-winning film in 1949.
Stevens will take the role of Morris Townsend, played by Montgomery Clift in the Hollywood drama.
Chastain will play Catherine Sloper - the part that won Olivia de Havilland her second Oscar for best actress.
The play, written by the husband and wife team Ruth and Augustus Goetz, tells of a plain young woman whose stern father suspects her latest suitor of wooing her for her inheritance.
David Strathairn will appear as the father, the role taken by Ralph Richardson on screen.
The Heiress, directed by Moises Kaufman, will open in October at a Broadway theatre yet to be announced.