None of Mr. Rosenstein’s proposals apparently came to fruition. It is not clear how determined he was about seeing them through, though he did tell Mr. McCabe that he might be able to persuade Attorney General Jeff Sessions and John F. Kelly, then the secretary of homeland security and now the White House chief of staff, to mount an effort to invoke the 25th Amendment.

The extreme suggestions show Mr. Rosenstein’s state of mind in the disorienting days that followed Mr. Comey’s dismissal. Sitting in on Mr. Trump’s interviews with prospective F.B.I. directors and facing attacks for his own role in Mr. Comey’s firing, Mr. Rosenstein had an up-close view of the tumult. Mr. Rosenstein appeared conflicted, regretful and emotional, according to people who spoke with him at the time.

Mr. Rosenstein disputed this account.

“The New York Times’s story is inaccurate and factually incorrect,” he said in a statement. “I will not further comment on a story based on anonymous sources who are obviously biased against the department and are advancing their own personal agenda. But let me be clear about this: Based on my personal dealings with the president, there is no basis to invoke the 25th Amendment.”

A Justice Department spokeswoman also provided a statement from a person who was present when Mr. Rosenstein proposed wearing a wire. The person, who would not be named, acknowledged the remark but said Mr. Rosenstein made it sarcastically.

But according to the others who described his comments, Mr. Rosenstein not only confirmed that he was serious about the idea but also followed up by suggesting that other F.B.I. officials who were interviewing to be the bureau’s director could also secretly record Mr. Trump.

Mr. McCabe, who was later fired from the F.B.I., declined to comment. His memos have been turned over to the special counsel, Robert S. Mueller III, in the investigation into whether Trump associates conspired with Russia’s election interference, according to a lawyer for Mr. McCabe. “A set of those memos remained at the F.B.I. at the time of his departure in late January 2018,” the lawyer, Michael R. Bromwich, said of his client. “He has no knowledge of how any member of the media obtained those memos.” (read more)