The deputy attorney general, Rod J. Rosenstein, suggested last year that he secretly record President Trump in the White House to expose the chaos consuming the administration, and he discussed recruiting cabinet members to invoke the 25th Amendment to remove Mr. Trump from office for being unfit.

Mr. Rosenstein made these suggestions in the spring of 2017 when Mr. Trump’s firing of James B. Comey as F.B.I. director plunged the White House into turmoil. Over the ensuing days, the president divulged classified intelligence to Russians in the Oval Office, and revelations emerged that Mr. Trump had asked Mr. Comey to pledge loyalty and end an investigation into a senior aide.

Mr. Rosenstein was just two weeks into his job. He had begun overseeing the Russia investigation and played a key role in the president’s dismissal of Mr. Comey by writing a memo critical of his handling of the Hillary Clinton email investigation. But Mr. Rosenstein was caught off guard when Mr. Trump cited the memo in the firing, and he began telling people that he feared he had been used.

Mr. Rosenstein made the remarks about secretly recording Mr. Trump and about the 25th Amendment in meetings and conversations with other Justice Department and F.B.I. officials. Several people described the episodes, insisting on anonymity to discuss internal deliberations. The people were briefed either on the events themselves or on memos written by F.B.I. officials, including Andrew G. McCabe, then the acting bureau director, that documented Mr. Rosenstein’s actions and comments.

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