A decade and a half ago, in the run-up to the invasion of Iraq, President George W. Bush’s administration conjured up not only terrifying images of nuclear mushroom clouds but also of Saddam Hussein plotting with Osama bin Laden to attack the United States.

Mr. Bush himself declared that Mr. Hussein “aids and protects terrorists, including members of Al Qaeda” while Secretary of Defense Donald Rumsfeld called links between Iraq and Al Qaeda “accurate and not debatable.”

It wasn’t true, of course. But it helped make the case for war.

That may be why a similar lie is getting trotted out again now, except this time the target is Iraq’s neighbor, Iran.

On Oct. 13, in his statement decertifying the Iran nuclear deal, President Trump claimed that Tehran “provides assistance to Al Qaeda.” The following week, his C.I.A. chief, Mike Pompeo, went further: “It’s an open secret and not classified information that there have been relationships, there are connections,” Mr. Pompeo said at an event hosted by the Foundation for Defense of Democracies, a neoconservative think tank. “There have been times the Iranians have worked alongside Al Qaeda.”

On Nov. 1, the C.I.A. released a new batch of nearly 470,000 files recovered in the 2011 raid on Osama bin Laden’s compound in Pakistan. But the agency did more than just release the documents to the public. It provided advance copies to the foundation’s online publication, Long War Journal. (The C.I.A. said it was common practice to distribute declassified documents to the news media and academic organizations on an embargoed basis and that the only agenda in releasing these files was “to enhance public understanding” of Al Qaeda.)

Long War Journal homed in on a 19-page document by an unidentified Qaeda official who claimed that the Iranian government had offered “Saudi brothers” in Al Qaeda “everything they needed,” including money, arms and training in Hezbollah camps in Lebanon, “in exchange for striking American interests in Saudi Arabia and the Gulf.” Yet the Iranian offer, admitted the official, was never accepted by Al Qaeda — if such an offer was in fact made.

The timing of these latest claims from the president and his C.I.A. chief are hardly coincidental. Tensions in the Middle East are ramping up. America’s chief allies in the region, Saudi Arabia and Israel, are pushing even more aggressively than usual to confront Iran. With the Obama administration gone, they have found a soul mate in the White House.

President Trump has staffed his administration with hawks who believe that the road to solving the Middle East’s problems runs through Tehran. Nikki Haley, the American ambassador to the United Nations, has accused Iran of trying to “hold the world hostage to its bad behavior.” Defense Secretary James Mattis once described the three biggest threats to American national security as “Iran, Iran, Iran.”

A former State Department official who worked under Mr. Trump’s secretary of state, Rex Tillerson, told me that the Trump administration is “obsessed” with Iran in the same way that the Reagan administration was obsessed with the Soviet Union. Inside the government, the former official added, “Iran, ISIS, Al Qaeda, are all mentioned in the same breath, as a menacing threat.”

But Americans aren’t exactly itching for a new war. (A majority, in fact, believes the country would be better off staying in the nuclear deal with Iran.) So how can the Trump administration build a case for a pre-emptive strike?

Those claims of a nefarious alliance with Al Qaeda might help. The “bomb Iran” crowd has long pointed to the presence of senior Qaeda officials, including members of the Bin Laden family, inside Iran since late 2001.

But Iran is far from being a base or command center for Al Qaeda. In 2001, after hundreds of Qaeda fighters crossed into Iran from Afghanistan fleeing American airstrikes, the Iranians deported most of them back to their countries of origin. In 2003, the Iranians offered to swap Qaeda members held under house arrest for members of Mujahedeen Khalq, a militant group that seeks to overthrow the Iranian government, who are being detained by American forces in Iraq.

The relationship between the Salafi Sunnis of Al Qaeda and the Shiite clerics of Iran is “not one of alliance” but “highly antagonistic” and “largely based on indirect and unpleasant negotiations over the release of detained jihadis and their families, including members of Bin Ladin’s family,” according to a 2012 report by the Combating Terrorism Center at the United States Military Academy at West Point. The report said that Iran held onto senior Qaeda figures not to protect or assist them but to use them as bargaining chips with the United States and also as a deterrent against Qaeda attacks.

When I asked terrorism experts what they made of the alleged Iran-Al Qaeda ties, they were unanimous in their incredulity.

“I’ve never seen any evidence of active collaboration,” said Jason Burke, the author of an acclaimed book on Al Qaeda.

Ali Soufan, a former F.B.I. agent and the author of the new book “Anatomy of Terror,” dismissed the coverage of the C.I.A.’s documents as an “oversimplification of the facts” and a result of “the Trump administration joining Saudi Arabia’s anti-Iran campaign.”

Few would deny that Iran has sponsored groups listed by the United States as “foreign terrorist organizations,” such as Hamas and Hezbollah. But, say the experts, support for Al Qaeda is another matter altogether. As William McCants, a former American government adviser on extremism and author of a recent book on the Islamic State, put it, Iran and Al Qaeda “never embraced as lovers.”

So far, none of the documents newly released by the C.I.A. contains a smoking gun. Have Iranian security forces and members of Al Qaeda had contacts, or done deals? Probably. Are there Qaeda figures still living in Iran? Almost definitely. Does that mean there’s an anti-American alliance between Iran and Al Qaeda? No.

Mr. Trump, like Mr. Bush before him, is beating the drum for war in the Middle East. But he needs a pretext for an attack on a sovereign nation that, according to the International Atomic Energy Agency, is complying with the terms of the nuclear deal. The American public fell for a false pretext in 2003 — and it cannot afford to do so again. Saddam Hussein was not allied with Al Qaeda; for all its faults, neither is the Iranian government.

As Mr. Bush himself once famously tried, yet failed, to say: “Fool me once, shame on you; fool me twice, shame on me.”