Top Republicans on Sunday called for the appointment of a new special counsel to examine the apparent deep-seated, anti-Trump animus inside the Justice Department, citing revelations that Deputy Attorney General Rod Rosenstein discussed secretly recording the president.

“Before the election, FBI officials tried to taint the election, tip it to [Hillary] Clinton’s favor. After the election, they’re trying to undermine the president,” Sen. Lindsey Graham told “Fox News Sunday.”

“If Rosenstein is involved, he should be fired. If he is not involved, leave him alone,” the South Carolina Republican said. “There’s a bureaucratic coup going on at the Department of Justice and the FBI, and somebody needs to look at it.”

Mr. Rosenstein in May 2017 reportedly suggested that he wear a wire and secretly record his conversations with the president. The deputy attorney general has strongly denied those reports. He reportedly also whispered to Cabinet officials about invoking the 25th Amendment to remove President Trump from office.

Amid the latest fears of a high-level “bureaucratic coup,” Mr. Trump is vowing to weed out the “lingering stench” of hostility toward him that he says exists inside the Justice Department. Mr. Rosenstein reportedly made the “wire” remark during a meeting with then-acting FBI Director Andrew McCabe, who took over after Mr. Trump dismissed Director James B. Comey and who thoroughly documented meetings with Mr. Rosenstein and other officials.

The president has described Mr. McCabe, who was later fired, as a liar.

Top Republicans say the episode is the latest chapter in a story that deserves its own investigation on a level equal to Robert Mueller’s ongoing Russia collusion probe.

Mr. Rosenstein’s alleged comments were first reported Friday by The New York Times, citing sources familiar with the conversations or with Mr. McCabe’s subsequent memos. The deputy attorney general strongly denied the report.

“The New York Times’s story is inaccurate and factually incorrect,” Mr. Rosenstein said in a statement. “I will not further comment on a story based on anonymous sources 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.”

“I never pursued or authorized recording the President and any suggestion that I have ever advocated for the removal of the President is absolutely false,” he said in a follow-up statement more forcefully denying the New York Times story.

A Justice Department spokesperson later said that the deputy attorney general did in fact mention wearing a wire but that he did so sarcastically and never intended to follow up his words with actions. The Justice Department also has drawn attention to other instances in which Mr. McCabe has made statements that later have been proved false.

The New York Times over the weekend stood by its reporting on the matter.

Leading Republican lawmakers say the president is unlikely to simply forget about the issue.

“Rod deserves the right to be heard, and I’m sure at some point the president will bring Rod in and say, ‘Rod, if you think I’m incompetent, if you feel the need to wear a wire when you talk to me, then why are you serving in the administration?’” Rep. Trey Gowdy, South Carolina Republican, told CBS’s “Face the Nation” program on Sunday. “Give Rod a chance to explain whether it’s true and the context in which it was said.”

Lawmakers also have said that they may issue subpoenas for Mr. McCabe’s memos, the contents of which consistently get leaked to media outlets but have not been provided to Congress.

The latest episode adds more fuel to the battle between Mr. Trump and the Justice Department. The president has consistently criticized his attorney general, Jeff Sessions, for recusing himself from the Russian collusion investigation. He also has held up anti-Trump texts from former FBI agent Peter Strzok and former FBI lawyer Lisa Page as proof that some within the agency wanted to ruin his presidency.

For months, there has been speculation that Mr. Trump could fire Mr. Sessions, Mr. Rosenstein or perhaps both. At a rally in Missouri on Friday night, the president said that “95 percent” of Justice Department and FBI employees support him, but “you’ve got some real bad ones” still inside.

“There’s a lingering stench, and we’re going to get rid of that, too,” Mr. Trump said.

The New York Times report also marks the second time in the past several weeks that the notion of using the 25th Amendment to remove Mr. Trump from office has come into the spotlight. An unidentified senior administration official said the Cabinet had discussed using that amendment — which can be used to oust the president if he is deemed unfit for office — in an anonymous New York Times op-ed this month.

Top administration officials responded to that op-ed and the revelations around Mr. Rosenstein in much the same way: telling anyone who believes the president is unfit for office to find a new job.

“I’m not going to comment on that in any way other than to say this: I’ve been pretty clear since my beginning of service here in this administration, if you can’t be on the team, if you’re not supporting this mission, then maybe you ought to find something else to do,” Secretary of State Mike Pompeo told “Fox News Sunday.”

“I’ve told that to my senior colleagues, I’ve told it to junior folks at the CIA and the State Department. We need everyone who’s engaged in helping achieve President Trump’s mission,” he said. “And I hope that everyone in every agency: DOJ, FBI, State Department is on that mission.”

Mr. Pompeo, who served as CIA director before being tapped as the nation’s top diplomat, also said he never heard any discussion among Cabinet officials about invoking the 25th Amendment.

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