Republican lawmakers Tuesday demanded that Deputy Attorney General Rod Rosenstein testify before Congress this week or face impeachment as questions swirl about his future at the Department of Justice.

House Freedom Caucus Chairman Rep. Mark Meadows said Mr. Rosenstein must explain under oath reports that he discussed secretly taping President Trump and advocated removing him from office.

“You can’t have the number two official at the Department of Justice making comments about wiring the president and not address it,” tweeted Mr. Meadows, North Carolina Republican. “Rod Rosenstein must come before Congress this week, under oath, and tell the truth about his alleged statements.”

Mr. Meadows is the second Republican member of Congress to threaten impeachment if Mr. Rosenstein does not testify.

“Any member of Congress can call up Rosenstein’s impeachment and it must be voted on in two days,” Rep. Matt Gaetz, Florida Republican, said in a Fox News interview. “If we don’t have Rosenstein in the witness chair, Mark Meadows and I are prepared to do just that and keep Congress in town to do the oversight.”

Mr. Rosenstein is scheduled to meet with Mr. Trump on Thursday to discuss a New York Times report saying he suggested recording the president and tried to recruit Cabinet members to invoke the 25th Amendment to remove him from office.

As Mr. Rosenstein headed to the White House on Monday to meet with Chief of Staff John F. Kelly, erroneous news reports said he had been fired or was going to resign.

He kept his job, though the White House remains mum about his future, with White House press secretary Sarah Huckabee Sanders dodging questions in television interviews Tuesday.

When asked on “Good Morning America” if the president has confidence in Mr. Rosenstein, Mrs. Sanders responded, “He has confidence in the system.”

Mrs. Sanders was pressed for greater detail but simply said she wasn’t “going to get ahead of the conversation” between the president and Mr. Rosenstein, the No. 2 official at the Justice Department.

During an appearance on “Fox & Friends,” Mrs. Sanders said the president is disappointed by a number of actions at the Justice Department. She said Mr. Trump wants answers from Justice officials, which could involve Mr. Rosenstein, but also others.

Democrats, meanwhile, are moving to protect special counsel Robert Mueller’s investigation into possible collusion between the Trump campaign and Russia. Mr. Rosenstein oversees the Mueller probe because Attorney General Jeff Sessions recused himself.

If Mr. Rosenstein is fired or resigns, it could heighten fears that Mr. Trump will terminate the Mueller investigation, which he has labeled a “hoax” and “witch hunt.”

Sen. Mark Warner of Virginia, the top Democrat on the Senate Intelligence Committee, told CNN on Monday a bill to protect the special counsel “ought to be passed right away.” Mr. Warner cautiously said the legislation has bipartisan support.

“There’s been a lot of my Republican colleagues along the way who have said if the time comes, they’ll step up. We will see later if that time comes along,” he told CNN.

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