The script is written by Lucas Hnath, whose play A Doll's House, Part 2, earned Metcalf her first Tony award.
Hillary and Clinton, directed by Joe Mantello, features the couple and campaign manager Mark J. Penn wrangling about how involved the former president should be in the campaign. Barack Obama is the play's unnamed fourth character, referred to in the script only as "The Other Guy."
Metcalf won her second Tony award earlier this year for her role in Edward Albee's Three Tall Women. Lithgow, currently performing John Lithgow: Stories by Heart on Broadway, has been nominated for six Tony awards and won twice.
Scott Rudin is slated to produce Hillary and Clinton, which is scheduled to begin previews March 16 and open April 18.