‘It’s not a movie about two gay guys,” Clint Eastwood barked when GQ asked him if his new movie, “J. Edgar,” out Wednesday, explored its title character’s mysteriously close relationship with his second-in-command.

Then again, Eastwood didn’t really give a definitive answer. And that’s what’s been keeping some J. Edgar Hoover defenders up at night ever since the project, starring Leonardo DiCaprio as the famously secretive founder of the Federal Bureau of Investigation, was announced.

Hoover’s intimate friendship with his deputy, Clyde Tolson, has been the subject of speculation for decades — Hoover died in 1972, Tolson three years later — but this is the highest-profile examination yet of the man who served under nine presidents and is best-known for his relentless persecution of supposed communists.

“I wrote Mr. Eastwood a letter,” says William Branon, chairman of the J. Edgar Hoover Foundation, “saying, please don’t do this. Mr. Hoover wasn’t a homosexual, nor did he have a homosexual relationship or engage in cross-dressing or any of that stuff.

“He responded immediately — two days later,” Branon adds. “He wrote a very satisfactory letter. He said he didn’t put any credence in those stories and wasn’t going to portray any homosexual relationship between Hoover and Tolson.”

Well, maybe not explicitly homosexual, but there is no mistaking the undercurrent of repressed affection in Eastwood’s depiction. When Hoover tells Tolson that he’s been seeing a woman for dinner and “it might be ‘time for a Mrs. Hoover’,” Tolson is infuriated.

“Don’t you make a fool of me, Edgar,” he says, precipitating a fight that culminates in the friends wrestling on the ground. Tolson sees that he has bloodied Hoover’s lip, then passionately kisses him with an open mouth.

“Don’t you ever do that again,” Hoover says when they pull apart.

“I won’t,” Tolson says.

But how close is this portrayal to the truth? Branon says not at all.

“If it was true [that they were involved],” Branon says, “it wouldn’t be the worst thing in the world. But it just isn’t. It’s not an accurate portrayal.” Branon, who worked for the FBI for 32 years, says Hoover wouldn’t have been able to keep a secret like that from a bureau filled with nosy investigative types.

“They’ve got naturally inquisitive minds,” he says. “If he was gay, it would have been all over the bureau. We would have known about it. For 20 years, he was under FBI surveillance for his own protection. They called it Hoover Watch. They would follow him from home to the office, to the Mayflower [Hotel] where he would go to eat lunch, to work, to home.”

Richard Hack, author of the 2004 biography “Puppetmaster: The Secret Life of J. Edgar Hoover,” agrees with Branon. “It just doesn’t fit with who Hoover was at all,” he says. “The man was so careful about his private life. If you’re trying to hide something, the last thing you’re going to do is appear in public every day with someone.”

Hoover merely prized Tolson’s steel-trap mind, Hack says. “He memorized everything. He’s the one Hoover would call and get the information from.”

He was also the only person in the FBI director’s life who was close to being a confidant. “Hoover basically had no friends,” Hack says. “He didn’t want friends. He didn’t want to be coerced or manipulated in any way, because he was the manipulator. With Tolson, he found a person who was devoted to the work that was his life. It was what he wanted.”

In addition to work, the two did vacation together. When Hoover died, he left his money to Tolson. And they are buried side by side in the Congressional Cemetery. For all of these reasons, some historians speculate there was more to the relationship.

“It is clear that Hoover never married, and that the most important relationship in his life was with Clyde Tolson,” says David K. Johnson, author of “The Lavender Scare: The Cold War Persecution of Gays and Lesbians in the Federal Government.”

“He may not have been homosexual, but he was not a conventional heterosexual either,” says Johnson. “Modern terms may not apply to men of his generation. His relationship with Tolson resembles a 19th-century romantic friendship, a common form of close, same-sex relationship that may or may not have been physical.”

It’s also clear that Hoover was deeply homophobic, in addition to his other prejudices: “He didn’t like Jews, he didn’t like blacks, he didn’t like homosexuals,” says Hack. “He hated everybody. And he would go after people.”

But it’s that very obsession that raises a red flag, some say. “We do have evidence from history of politicians hiding their sexual orientation behind anti-gay posturing,” says Nathaniel Frank, author of “Unfriendly Fire: How the Gay Ban Undermines the Military and Weakens America.” “That history, along with logic, suggests that being anti-gay [is] a good way to hide [the fact you’re] gay.”

We may never know about Hoover, which Eastwood hints at in an enigmatic final scene involving the FBI boss’ own dossier.

“If the film hints at a very close, ambiguous relationship with Tolson,” says Johnson. “That would be about as close to the truth as we can get.”