Neil Diamond attended a very private run-through of a Broadway-bound musical about his life last week — and was spotted humming along to his songs. The show, as yet untitled, is now on track to hit New York in August 2021 after an out-of-town tryout.

Broadway’s been inundated with bio musicals about pop icons, from Tina Turner to Cher, Carole King to Donna Summer. “Ain’t Too Proud,” the musical about the Temptations, is grossing nearly $2 million a week.

But the Diamond show may stand out because of its writer — Anthony McCarten, who’s up for an Oscar for his movie “The Two Popes.” McCarten also wrote “The Theory of Everything” (about Stephen Hawking), “Darkest Hour” (about Winston Churchill’s early days as prime minister) and the Queen movie, “Bohemian Rhapsody.”

McCarten is no stranger to theater. He wrote 12 plays before becoming one of the hottest screenwriters in Hollywood.

A huge Diamond fan, the New Zealand native spent some time with the singer last year soaking up the details of his life. The musical charts his rise from poverty in Brooklyn to international stardom. Sources say it doesn’t shy away from the 79-year-old singer’s personal problems, including issues with his two failed marriages and struggles with depression. It also deals with Diamond’s time at the Brill Building, where he wrote “I’m a Believer,” which went gold after just two days and topped the charts for seven weeks.

The score features several other Diamond classics: “I Am . . . I Said,” “Forever in Blue Jeans,” “Song Sung Blue,” “America” and more.

“The hardest part of developing the show is that the catalog is so big and fantastic,” says a production source. “Ten No. 1 hits — how do we include them all? But you can bet audiences will hear ‘Sweet Caroline.’ ”

Ba, ba, ba!

Diamond retired from touring two years ago after being diagnosed with Parkinson’s disease. I saw what I believe was his last public performance in June 2018 at the Songwriters Hall of Fame. He received the Lifetime Achievement Award, and was in sensational form onstage.

Diamond was “visibly moved” at last week’s run-through, says a source. And he was delighted by Will Swenson, who played him. Swenson’s Broadway credits include “Hair,” “Priscilla Queen of the Desert” and “Waitress.”

Michael Mayer, a Tony winner for “Spring Awakening,” is directing the show. Ken Davenport and Bob Gaudio are producing. Gaudio knows a thing or two about bio musicals: “Jersey Boys,” based on Gaudio’s old band, the Four Seasons, ran 12 years on Broadway and grossed more than $2 billion worldwide before working its way to off-Broadway, where it’s still running,

Between McCarten and Gaudio, Diamond should be in good hands.

All of Broadway will come together Feb. 3 for the Jerry Herman tribute at the Lunt-Fontanne. Produced by Michael Feinstein, the lineup of performers includes Sutton Foster, Kelli O’Hara, Kristin Chenoweth and Marilyn Maye. Harvey Fierstein is speaking, and I know he’s got great stories about working with Herman, who died Dec. 26, on “La Cage aux Folles.”

Here’s my favorite: At the first public performance of “La Cage” at Boston’s Colonial Theatre, Gene Barry forgot some of Herman’s lyrics to “Song on the Sand.”

“A fellow with a concertina sang,” Barry crooned, “what was the song? It’s strange what we recall, and odd what we forget. I heard — ”

Then he froze.

George Hearn took his hand and sang the lyric: “La da da da da da da . . . ”

“Len Berman and Michael Riedel in the Morning” airs weekdays on WOR Radio 710.