It is an arresting, menacing image: the scarred face of Roy M. Cohn, the notorious New York City lawyer, fixer and master of the political dark arts, floating bodiless in a pool of black, his hooded eyes staring directly back at the viewer.

The photograph, taken by Robert Mapplethorpe in 1981, is an unforgettable depiction of its subject, and the celebrated author Kai Bird thought it would make the perfect cover art for his new biography of Mr. Cohn, “American Scoundrel,” which will be released by Scribner next week.

That the picture never ended up on the book’s cover is a matter of regret to Mr. Bird — and a small but telling illustration of the anxiety that has taken root in the nonprofit world because of Mr. Cohn’s most famed protégé, an ambitious real estate developer now serving his second term as president.

President Trump loomed large in the decision by the Robert Mapplethorpe Foundation, which administers the artist’s estate and catalog, to decline the request to license the photograph, according to three people familiar with its deliberations.

Mr. Mapplethorpe, who died in 1989, is widely remembered for fighting censorship. In this case, the foundation’s board feared that Mr. Trump could retaliate, perhaps by threatening its nonprofit tax status, if the president — who has used the levers of government to pressure universities, law firms and TV networks — were to take offense, the people said.

As a charitable organization, the Mapplethorpe Foundation is legally prohibited from engaging in certain political activities. Mr. Bird’s book is subtitled “Roy Cohn’s Dark Journey From Joe McCarthy to Donald Trump,” and the board was concerned that the photo’s inclusion could be construed as a kind of political act, the people said.

It would not be the first time that a cultural organization has been alarmed by the president’s hardball tactics — some of which he honed under Mr. Cohn, who prosecuted Julius and Ethel Rosenberg and aided Senator Joseph McCarthy in his witch hunt for communists.

Mr. Trump has demanded changes at the Smithsonian and taken control of the John F. Kennedy Center for the Performing Arts. He has threatened to strip Harvard of its tax-exempt status. Nonprofit groups, like the Southern Poverty Law Center and the Democratic fund-raising platform ActBlue, have been investigated by his Justice Department.

Mr. Bird, whose 2005 biography of Robert Oppenheimer, “American Prometheus,” was adapted into the Oscar-winning Christopher Nolan film, said in an interview that he was “thunderstruck” and “outraged” when he learned that the Mapplethorpe Foundation had declined to license the portrait. “It’s astonishing that these people are being so cowed, so easily,” he said.

In his biography of Mr. Cohn, Mr. Bird writes that the foundation’s decision was “yet another example of Roy’s ghost haunting us from the grave.”

The Mapplethorpe Foundation declined to comment on Mr. Bird’s criticisms.

Mr. Mapplethorpe started the foundation shortly before his death “to protect his work, to advance his creative vision and to promote the causes he cared about,” according to the foundation’s website.

His photography, which explicitly depicted the sexual lives of gay men, was the subject of several well-publicized battles over free expression, notably when Jesse Helms, the conservative North Carolina senator, objected in 1989 to the use of federal funds for exhibits of his work.

When the matter of Mr. Bird’s biography arose last year, the artist’s foundation made a different calculus. Its board determined that any kind of government retaliation could drain its financial resources, creating an existential risk for the organization, according to the people familiar with the board’s decision-making. And while the biography may have brought attention to Mr. Mapplethorpe’s work, the photo would ultimately be in service of one author’s commercial endeavor.

Several First Amendment lawyers, none of whom were involved in the foundation’s discussions, said the use of a photograph for a scholarly biography would not typically constitute political activity. But the lawyers acknowledged that it was difficult to ignore the uneasy atmosphere the Trump administration has created.

“The idea that a photograph of a dead mobster lawyer could somehow qualify as political activity, even under the Trump administration, is hard to fathom,” said Suzanne Nossel, a former chief executive of the free expression group PEN America. “And yet, we’re in an environment where no one feels they can rule anything out.”

The photograph itself has a colorful history.

In 1981, Andy Warhol dispatched Steven M.L. Aronson, a writer and book editor, to conduct a Q. and A. with Mr. Cohn for an issue of Interview magazine, Mr. Warhol’s arts and culture publication. By then, Mr. Cohn was a New York celebrity: a power player who partied at Studio 54 and represented moguls, mobsters and media barons.

Mr. Aronson persuaded Mr. Cohn to pose for Mr. Mapplethorpe, a friend whose work was still relatively obscure. Their meeting was an unusual encounter between an artist who celebrated the lives of gay men and a closeted lawyer whose anti-gay virulence would later be depicted in the Tony Kushner play “Angels in America.” Both Mr. Cohn and Mr. Mapplethorpe would die from AIDS complications before the decade’s end.

Mr. Cohn adored the photograph of himself, and later dined with Mr. Mapplethorpe and Mr. Aronson at the downtown New York restaurant Raoul’s. “He loved publicity, for one thing,” Mr. Aronson recalled of Mr. Cohn, “and the photograph was glamorous, as evil often is.”

For the cover of “American Scoundrel,” Mr. Bird and his publisher ultimately settled on a portrait by the photographer Britain Hill, showing Mr. Cohn in a blue pinstriped suit with his hands folded beneath his chin.

He is still staring directly at the viewer.