Rod Rosenstein remains the No. 2 man at the Justice Department, but now faces a major meeting later this week with President Trump to hash out his future.

News reports that Mr. Rosenstein had resigned or was fired Monday turned out not to be true.

White House press secretary Sarah Huckabee Sanders said Mr. Trump does want to have a face-to-face meeting with his deputy attorney general to talk about allegations Mr. Rosenstein talked about secretly taping the president, and suggested invoking the 25th Amendment to oust the president from office.

Mr. Rosenstein on Monday did have a previously scheduled meeting with White House Chief of Staff John F. Kelly. As Mr. Rosenstein departed a White House meeting, Mr. Kelly took the unusual step of giving the smiling Mr. Rosenstein an enthusiastic goodbye handshake on the driveway in full view of television cameras, as if to convey that their discussion was amicable.

In New York for a meeting at the United Nations, Mr. Trump confirmed that he spoke with Mr. Rosenstein on Monday and he still has his job at the Justice Department. When asked if he was firing Mr. Rosenstein, the president responded, “no,” adding that the pair will have a meeting on Thursday.

“We’ll be meeting at the White House and we will be determining what’s going on,” Mr. Trump said. “We want to have transparency. We want to have openness and I look forward to meeting with Rod at that time.”

Mr. Rosenstein has denied suggesting removing the president, though a department official told The New York Times he did — sarcastically — mention taping Mr. Trump.

The explanations and denials have done little to stop calls from conservatives for Mr. Rosenstein to step down or be fired if he’s been undermining the president.

“If Rosenstein is involved, he should be fired,” said Sen. Lindsey Graham, South Carolina Republican. “There’s a bureaucratic coup at the Department of Justice and FBI, and somebody needs to look at it.”

Democrats warned against any moves to oust the deputy attorney general, who is overseeing special counsel’s Robert Mueller’s probe into Russian meddling in the 2016 election.

“Under no circumstances should Rod Rosenstein resign,” tweeted Rep. Adam Schiff, the top Democrat on the House intelligence committee. “This would place the Mueller investigation in even greater jeopardy. Rosenstein should continue to do his job, protect the independence of the DOJ, and if the president intends to obstruct justice, force Trump to fire him.”

Other Democrats said it doesn’t matter whether Mr. Rosenstein is fired or resigns — either would be a hindrance to the Mueller probe.

Rep. Elijah Cummings, the ranking Democrat on the House Oversight Committee, said his panel must be ready to hold a hearing to get evidence from Mr. Rosenstein.

“Mr. Rosenstein’s removal would plunge our nation into uncharted territory and pose a serious and profound threat to the continued work of the special counsel, and I would expect the American people to be outraged if President Trump’s extended campaign to interfere with this investigation results in Mr. Rosenstein’s ouster,” he said.

Conservative lawmakers said The New York Times report underscores the need for Mr. Trump to declassify Foreign Intelligence Surveillance Act records related to former campaign official Carter Page. Mr. Trump was set to release the documents despite Justice Department concerns they will have a negative impact on the Mueller probe. He backed off releasing everything last week.

“The latest reports on Rod Rosenstein underscore the desperate need for transparency at the DOJ,” tweeted Rep. Mark Meadows, North Carolina Republican. “Release the documents. Declassify everything. Stop the games and show Americans the truth about this Russia investigation.”

Attorney General Jeff Sessions recused himself from the investigation into Russian meddling, leaving Mr. Rosenstein in charge of the investigation.

Should Mr. Rosenstein go, the next in line would be the No. 3 spot at the Justice Department, but it’s been vacant since Associate Attorney General Rachel Brand resigned earlier this year.

Under the Justice Department’s succession plan, Solicitor General Noel Francisco would be next in line.

But some analysts say he might not be the best choice. Mr. Francisco has said little about the special counsel investigation, but has suggested federal political corruption investigations have overreached.

He also accused former FBI Director James B. Comey of using “kid gloves” in the investigation into Hillary Clinton’s use of a private email server while secretary of state.

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