When an aging movie goddess, Ava Gardner, asks a journalist to write her memoir in Elizabeth McGovern’s new play, “Ava: The Secret Conversations,” she is blunt about her motives: “I need the money.” Gardner is portrayed in this one-act piece’s off-Broadway premiere, part of a North American tour, by Ms. McGovern, who is better known as an actress herself.

The latter-day screen star — a popular film ingenue in the 1980s, more recently known for her work in the “Downton Abbey” series and movie adaptations — based this play on the book “Ava Gardner: The Secret Conversations,” culled from interviews conducted by a British reporter and author, Peter Evans, and credited to both him and Gardner when it was published in 2013, the year after his death.

Gardner, who died in 1990, had begun her discussions with Evans a couple of years before that, but eventually fired the writer, who as it turned out had been sued for libel by one of her three ex-husbands, Frank Sinatra. An autobiography organized with other collaborators, “Ava: My Story,” was released shortly after her passing, but Evans’s book, completed after he had finally secured the rights later in his life, is not surprisingly considered the juicier and more compelling of the two.

It’s not hard to see why Ms. McGovern was drawn to Gardner’s story. The playwright and leading lady’s dark-haired, ivory-skinned beauty recalls her subject somewhat, though Ms. McGovern, who is now in her 60s, as Ava is when the play unfolds, looks far more fit than the late actress did even in early middle age. Her hard living was the stuff of legend. 

Yet if Ms. McGovern’s own life has not, to my knowledge, been as turbulent as Gardner’s apparently was, she likely, as a gorgeous young woman seeking her fortune in the film industry — even at a slightly more enlightened time — encountered certain pressures and assumptions similar to those Gardner faced.

These challenges are well documented in the play, in which Evans is also represented: Ms. Gardner’s “Conversations” is by turns a debate and a dance of seduction between the two. Peter is played by Aaron Costa Ganis, a stage and screen veteran, and between his scenes with Ava, which unfold in the London apartment where Gardner spent her final years — handsomely appointed here by scenic designer David Meyer — we hear the writer’s tortured exchanges with his literary agent, Ed Victor, who reminds his client repeatedly and in the crudest of terms what his task should be.

“The penis, Peter!” Ed bellows, unseen but voiced with the right crass urgency by John Tufts, referring to Sinatra’s. While Peter has illusions of being the next F. Scott Fitzgerald, Ed keeps prodding him to focus on matters of more lascivious interest, principally Ava’s tempestuous relationship with her third husband and her two previous marriages, to a notoriously randy young Mickey Rooney and the bandleader Artie Shaw.

Ava, of course, would rather steer clear of this territory, but Peter disarms her by reading up on her exes and then, in segments that blur the line between fantasy and reality, morphing into them. (Alex Basco Koch’s vivid projection design, showcasing vintage clips of Gardner and her partners, adds color and context.) Inevitably, romantic tension develops between the married author and his lonely subject.

Under Moritz von Stuelpnagel’s brisk direction, Mr. Ganis impressively channels the famous, disparate men who inform this process, aping Rooney’s boyish eagerness and even offering a passable simulation of a mid-career Sinatra at the mic. The actor also manages to bring a low-key charm to the more prosaically written Peter, defined here as a frustrated novelist and a decent family man who’s nonetheless not immune to temptation.

The play is most effective as a vehicle for its creator. Ms. McGovern’s performance is immaculate in its detail, from her alternately slinky and strained movements, evoking Ava at different stages of life, to the way her ultra-polished, almost stilted pronunciation — the product of elocution lessons provided when the North Carolina-born actress first arrived at Hollywood — can slide back into a Southern drawl when conversation turns to her youth, or when she’s piqued.

“All my life I was the woman men dream about,” Ava tells Peter toward the end. “That was the only job I ever had. Do you see where it’s left me?” Ultimately, “Ava: The Secret Conversations” is as much a cautionary tale as a tribute; in both respects, Ms. McGovern’s play is, in its simple way, passionately drawn and engagingly delivered.