With Tony Awards season in full swing, the New York theater industry probably hasn’t taken much notice yet of “Josephine,” the new musical about Josephine Baker that opened in Florida on May 6 with its sights on Broadway.

New Yorkers may not know much about the project, but they’ll certainly recognize the producer: Kenneth Waissman, who in the 1970s shepherded a little musical called “Grease” to Broadway, and went on to produce plays including “Agnes of God” and “Torch Song Trilogy.” The industry veteran, who hasn’t had a Broadway outing since the short-lived 1997 soul-music revue “Street Corner Symphony,” has been developing “Josephine” for more than decade.

“I’ve always waited until I find something that really turns me on and makes me passionate about it,” Waissman said of the gap between projects. What grabbed him about the story of Josephine Baker — whom he met, back during her final Broadway appearance in 1973 — was her WWII-era stint in Paris, encompassing her work with the French Resistance and her romance with the Crown Prince of Sweden.

Waissman has corralled a creative team that includes playwright Mark Hampton (“Full Gallop”) and pop songsmiths Steve Dorff and John Bettis, with Joey McKneely (“Smokey Joe’s Cafe,” the 2009 revival of “West Side Story”) on board to direct and choreograph. Starring Deborah Cox (“Jekyll and Hyde,” “Aida”), “Josephine” launched its pre-Broadway run in late April in Sarasota, Fla. at the Asolo Repertory Theater, the theater where Frank Wildhorn musical “Bonnie and Clyde” had its pre-Broadway tryout in 2010.

Local arts coverage is a little thin on the ground in Sarasota, so reviews aren’t plentiful. But the two most prominent notices that have hit see promise in the project, while agreeing that more work needs to be done to make Josephine more relatable. “It’s criticism we took to heart,” Waissman said.

Next stop: Broadway, ideally sometime during this coming season. Waissman estimated that before the Asolo run had even begun he’d raised some 40% of the capitalization costs, initially pegged at $13.5 million. (He now thinks he can do it for less.)

It’s been a while since the producer has brought a project to Broadway, which over the last decade has scaled up into a billion-dollar business. “The principles, the basics, stay the same,” he said. “Only the numbers change. I remember when most musicals cost $500,000.”

“Josephine” wraps up its run at Asolo Rep May 29, with the exact details and timeline for its future life still to be set.