The F.B.I. director, Kash Patel, the director of national intelligence, Tulsi Gabbard, and others told employees not to respond to a directive from Mr. Musk to summarize their accomplishments.

Leaders at defense and intelligence offices were particularly wary of Musk’s demand over the weekend asking workers to outline what they did last week.

President Donald Trump backed Elon Musk’s demand that federal employees explain their recent accomplishments by the end of Monday or risk getting fired, even as government agency officials were told that compliance with Musk’s edict was voluntary.