Several Trump-appointed agency leaders urged federal workers not to comply with Elon Musk’s order to summarize their accomplishments for the past week or be removed from their positions, even as Mr. Musk doubled down on his demand over the weekend.

Their instructions in effect countermanded the order of Mr. Musk across much of the government, challenging the broad authority President Trump has given the world’s richest man to make drastic changes to the federal bureaucracy. The standoff serves as one of the first significant tests of how far Mr. Musk’s power will extend.

As the directive ricocheted across the federal government, officials at some agencies, including the F.B.I., the office coordinating America’s intelligence agencies and the Departments of Defense, State, Energy, Health and Human Services and Homeland Security, told their employees not to respond.

Mr. Musk’s email had even reached the inboxes of sitting federal judges — who are in the judicial branch, not the executive. The administrative office for the federal courts advised judges and staff that “this email did not originate from the judiciary or the administrative office and we suggest that no action be taken.”

The public pushback reflects a growing unease — and, in some cases, alarm — behind the scenes across the Trump administration about the perception of Mr. Musk’s unchecked power.

The unease runs from lower staff to some cabinet secretaries, who have tired of having to justify specific intricacies of agency policy and having to scramble to address unforeseen controversies that Mr. Musk has ignited.

Those officials are aware that he has influence over the president privately, and they fear him using X, the social media website he owns, to single out people he views as obstructing him, according to one senior administration official.

Hours after a senior Defense Department official publicly and firmly pushed back on Mr. Musk’s directive on Sunday afternoon, Mr. Musk singled him out for retribution, saying on X that “anyone with the attitude of that Pentagon official needs to look for a new job.”

One person who was quiet about the controversy throughout much of the weekend was Mr. Trump; after posting on social media on Saturday morning that he wanted Mr. Musk to be more “aggressive,” and then bragging about the purge of federal workers in a speech hours later, the president had remained mute on the subject for much of Sunday.

That afternoon, however, Mr. Trump posted a meme, which he said came from Mr. Musk, mocking federal workers who had to explain their duties and accomplishments, but he did not weigh in on the internal government conflict between his appointees.

Mr. Musk’s public statements about his cost-cutting effort, known as the Department of Government Efficiency, have often expressed an open contempt for the federal work force, which includes some of Mr. Trump’s supporters.

By Sunday afternoon, some of the pushback against Mr. Musk from administration officials — coming in large part from the national security apparatus and law enforcement agencies — had become public and explicit.

“The Department of Defense is responsible for reviewing the performance of its personnel and it will conduct any review in accordance with its own procedures,” Darin S. Selnick, the acting Pentagon official in charge of personnel, said in a statement, instructing Pentagon employees to “for now, please pause any response.”

Tulsi Gabbard, the director of the office of national intelligence, ordered all intelligence community officers not to respond, in a message to intelligence officials reviewed by The New York Times.

“Given the inherently sensitive and classified nature of our work, I.C. employees should not respond to the OPM email,” Ms. Gabbard wrote.

Kash Patel, the F.B.I. director, wrote in an email to employees that “the F.B.I., through the office of the director, is in charge of all our review processes,” telling workers that they should “for now, please pause any responses.”

Senior personnel officials at the State and Homeland Security Departments also instructed their employees to not respond to the email.

At the Justice Department and F.B.I., the threatening signals from Mr. Musk were met with a mix of anger and amazement that anyone would issue such a blanket demand without consideration for sensitive areas such as criminal investigations, legal confidentiality or grand jury material.

Some law enforcement supervisors quickly told employees to wait for more guidance from managers on Monday before responding to the demand, according to current and former officials.

Other departments gave conflicting guidance. The Department of Health and Human Services told its employees on Sunday morning to follow the directive. An hour later, an email from the Trump-appointed acting director of the National Institutes of Health, a subordinate agency, told employees to hold off on responding. Hours later, the health department told all employees to “pause” responses to the ultimatum.

On Saturday, Mr. Musk posted a demand for government employees to summarize their accomplishments for the week, warning that failure to do so would be taken as a resignation. Soon after, the Office of Personnel Management, which manages the federal work force, sent an email asking civil servants for a list of accomplishments, but it did not include the threat of removal for not complying.

Unions representing federal workers suggested that Mr. Musk’s order was not valid. They advised their members to follow guidance from their supervisors on how, and whether, to respond to the email.

In a scathing letter on Sunday, Everett B. Kelley, the president of the American Federation of Government Employees — the largest federal employee union — told the acting director of the Office of Personnel Management that the email sent to federal employees was “plainly unlawful” and “thoughtless.”

Mr. Kelley demanded that the order be retracted, and noted, “By allowing the unelected and unhinged Elon Musk to dictate O.P.M.’s actions, you have demonstrated a lack of regard for the integrity of federal employees and their critical work.”

Multiple intelligence agencies, including the National Security Agency, had warned employees that responding could risk inadvertently disclosing classified work.

Although Mr. Musk’s original email told employees not to include classified material, current and former intelligence officials said that if an adversary gained access to thousands of unclassified accounts of intelligence officers’ work that it would be able to piece together sensitive details or learn about projects that were supposed to remain secret.

Representative Mike Lawler, a New York Republican whose seat may be among the most fiercely contested in 2026, raised doubt about the order even as he gave broader support to Mr. Musk’s cost-cutting effort.

“I don’t know how that’s necessarily feasible,” Mr. Lawler said of the ultimatum. “Obviously, a lot of federal employees are under union contract.”

Senator Lisa Murkowski, Republican of Alaska, also criticized Mr. Musk’s order.

“Our public workforce deserves to be treated with dignity and respect for the unheralded jobs they perform,” she wrote in a statement on social media. “The absurd weekend email to justify their existence wasn’t it.”

It is unclear what legal basis Mr. Musk would have to justify mass firings based on responses to the email, and the White House and the Office of Personnel Management did not immediately answer questions about the threat of removal.

But Mr. Musk — who made similar unconventional demands during his takeover of Twitter, now known as X — insisted on Sunday morning that the order amounted to “a very basic pulse check.”

In a series of posts, Mr. Musk also promoted baseless claims of wage fraud — that a significant number of “non-existent” or dead people were employed in the federal work force, and that criminals were using the fake employees to collect government paychecks.

“They are covering immense fraud,” Mr. Musk said in response to a post by a supporter that said that “the left is flipping out about a simple email.”

His claims echo a similar one that tens of millions of dead people may be receiving fraudulent Social Security payments. A recent report by the Social Security Administration’s inspector general — a watchdog that investigates the program for waste, fraud and abuse — found that “almost none” of the people in the agency’s database who had likely died were receiving payments.

Reporting was contributed by Julian E. Barnes, Hamed Aleaziz, Apoorva Mandavilli, Devlin Barrett, Rebecca Davis O’Brien, Ken Bensinger, Kate Conger, Sheryl Gay Stolberg, Adam Goldman, Minho Kim, Kate Zernike, Lisa Friedman and Margot Sanger-Katz.