In the East Room of the White House on Saturday night, the director of the F.B.I., Kash Patel, was huddling with Tulsi Gabbard, the director of national intelligence.

It was just a few hours after Elon Musk threatened to fire federal workers who failed to send an email describing what they did during the workweek. The two cabinet officials found a quiet spot to confer on the sidelines of a dinner for the nation’s governors.

Obeying Mr. Musk’s order could reveal national security secrets and other sensitive information, they agreed, according to three people familiar with the conversation. Mr. Patel and Ms. Gabbard were still wholeheartedly behind Mr. Musk’s goal of slashing the size of the federal work force. But they could not allow their workers to follow the directive, even if it would defy Mr. Musk — and, more important, annoy the president.

Across the top ranks of the government, there were more conversations. Ms. Gabbard spoke with John Ratcliffe, the C.I.A. director. Chiefs of staff and senior personnel officers at the Defense Department and other agencies consulted one another about what to do, even as managers around the country fielded panicked questions from employees about whether they were at risk of being fired from their jobs.

Cabinet secretaries at major agencies found themselves struggling to figure out how to respond to a directive from the president’s most powerful adviser without compromising their own staff. Several secretaries and top staff at major agencies that deal with sensitive matters called the White House repeatedly over the weekend looking for guidance, according to people with knowledge of the matter.

Mr. Patel was the first to push back publicly.

“The F.B.I., through the Office of the Director, is in charge of all of our review processes, and will conduct reviews in accordance with F.B.I. procedures,” Mr. Patel wrote in an email to the bureau. “When and if further information is required, we will coordinate the responses. For now, please pause any responses.”