Stephen Frears directs the film about the opera singer, who was also the basis for the Broadway play-with-music Souvenir, which starred Tony winner Judy Kaye.
Grant is cast as Jenkins' manager and partner St. Clair Bayfield.
"Florence" features a screenplay by Nicholas Martin; a release date has yet to be announced.
Jenkins, who was an eccentric society woman, believed she was a great soprano, although the opposite was true. Despite her lack of talent, Jenkins' charity recitals in such venues as the Ritz Carlton Hotel brought her much fame. Jenkins often mistook audiences' muffled laughter for cheers, and over 2,000 people were turned away from her Carnegie Hall concert in the mid 1940's.
Streep was seen this past winter as The Witch in the film version of Stephen Sondheim and James Lapine's Into the Woods.