Sitting in the back of a performance studio in New York’s theater district, Huey Lewis watched the cast of “The Heart of Rock & Roll” perform the show in a recent, private presentation for Broadway theater owners. The musical, built around the hit songs of Huey Lewis and the News, had a successful six-week run at San Diego’s prestigious Old Globe theater last fall. Now it’s auditioning to get a venue in New York and open on Broadway.

Mr. Lewis, in jeans and a zip-up sweater, stomped his boots and played air saxophone, and...