Mr. Trump, the officials warn, “would be the most reckless president in American history.”

The letter says Mr. Trump would weaken the United States’ moral authority and questions his knowledge of and belief in the Constitution. It says he has “demonstrated repeatedly that he has little understanding” of the nation’s “vital national interests, its complex diplomatic challenges, its indispensable alliances and the democratic values” on which American policy should be based. And it laments that “Mr. Trump has shown no interest in educating himself.”

“None of us will vote for Donald Trump,” the letter states, though it notes later that many Americans “have doubts about Hillary Clinton, as do many of us.”

Among the most prominent signatories are Michael V. Hayden, a former director of both the C.I.A. and the National Security Agency; John D. Negroponte, who served as the first director of national intelligence and then deputy secretary of state; and Robert B. Zoellick, another former deputy secretary of state, United States trade representive and, until 2012, president of the World Bank. Two former secretaries of homeland security, Tom Ridge and Michael Chertoff, also signed, as did Eric S. Edelman, who served as Vice President Dick Cheney’s national security adviser and as a top aide to Robert M. Gates when he was secretary of defense.

Robert Blackwill and James Jeffrey, two key strategists in Mr. Bush’s National Security Council, and William H. Taft IV, a former deputy secretary of defense and ambassador to NATO, also signed.

The letter underscores the continuing rupture in the Republican Party, but particularly within its national security establishment. Many of those signing it had declined to add their names to a similar open letter released in March. But a number said in recent interviews that they changed their minds once they heard Mr. Trump invite Russia to hack into Mrs. Clinton’s email server — a sarcastic remark, he said later — and say that he would check to see how much NATO members contributed to the alliance before sending forces to help stave off a Russian attack.

Yet the signatories are unlikely to impress Mr. Trump or the largely lesser-known foreign policy team he has assembled around him: He has said throughout his campaign that he intends to upend Republican foreign policy orthodoxy on everything from trade to Russia. And many of the aides who signed the letter were active in developing the plan to invade Iraq or managing its aftermath, which Mr. Trump has described as a “disaster.”

A spokeswoman for Mr. Trump did not immediately respond to a request for comment.

Missing from the signatories are any of the living Republican former secretaries of state: Henry Kissinger, George P. Shultz, James A. Baker III, Colin L. Powell and Condoleezza Rice.

Mr. Trump met with Mr. Kissinger and Mr. Baker several months ago, and “I came away with a lot of knowledge,” he told The New York Times in a July 20 interview. But neither of the two — who represent different foreign policy approaches within the party — has said if he will endorse Mr. Trump.

It is unclear whether the former secretaries plan to stay silent or will issue their own statements. But particularly striking is how many of Ms. Rice’s closest aides at the White House and the State Department, including Philip Zelikow, Eliot A. Cohen, Meghan O’Sullivan, Kori Schake and Michael Green, are all signatories.

“We agreed to focus on Trump’s fitness to be president, not his substantive positions,” said John B. Bellinger III, who served as Ms. Rice’s legal adviser at the National Security Council and the State Department, and who drafted the letter.

Mr. Bellinger said that among the signatories, “some will vote for” Mrs. Clinton, “and some will not vote, but all agree Trump is not qualified and would be dangerous.”

The Clinton campaign appeared to be aware that the letter was circulating and encouraged it, but played no role in drafting it, several signatories said.

Yet perhaps most striking about the letter is the degree to which it echoes Mrs. Clinton’s main argument about her rival: that his temperament makes him unsuitable for the job, and that he should not be entrusted with the control of nuclear weapons.

“He is unable or unwilling to separate truth from falsehood,” the letter says. “He does not encourage conflicting views. He lacks self-control and acts impetuously. He cannot tolerate personal criticism. He has alarmed our closest allies with his erratic behavior. All of these are dangerous qualities in an individual who aspires to be president and commander in chief, with command of the U.S. nuclear arsenal.”

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