Deputy attorney general Rod Rosenstein remains in his job, after reports on Monday said he had resigned or offered to do so.

The White House said Donald Trump and Rosenstein had “an extended conversation” about news stories published last week reporting that Rosenstein discussed with colleagues secretly recording Trump or having him removed from office.

The two plan to meet on Thursday, after the president’s return from a United Nations general assembly meeting in New York, press secretary Sarah Sanders said in a statement. Sanders said the meeting was at Rosenstein’s request.

The White House did not comment on reports Rosenstein expected to be fired imminently or had resigned.

The deputy attorney general oversees the investigation of Russian election interference and links between the Trump campaign and Moscow.

The New York Times reported last week that he discussed secretly recording the president and invoking the 25th amendment to remove Trump from office. Rosenstein said the report was “inaccurate”.

On Monday morning, as reports of Rosenstein leaving his post spread, a spokesperson for the Department of Justice declined to comment.

If Rosenstein were to leave his job, it would throw into question the oversight of special counsel Robert Mueller’s investigation. Under the Department of Justice succession statute, solicitor general Noel Francisco would be left in charge of supervising Mueller, University of Texas law school professor Steve Vladeck has written.

But Trump also might be able to appoint a temporary replacement on an “acting” basis for up to 210 days, Vladeck writes – that person would need to have passed Senate confirmation for another post or be a senior official from a relevant department.

As anticipation of Rosenstein’s departure grew on Monday, Democrats in Congress and former federal prosecutors called on Republicans to speak out in favor of protecting the Mueller investigation.

“Congress must take immediate steps,” said Representative Val Demings of Florida in a statement.

“Time to protect the Mueller investigation. Now,” tweeted Preet Bharara, former US attorney for the southern district of New York.

Mueller’s office declined to comment.

Rosenstein was overseeing Mueller’s work after the attorney general, Jeff Sessions, recused himself due to his contacts with Russian officials when he was part of the Trump campaign. Trump has repeatedly attacked Sessions for that move.

In overseeing the special counsel, Rosenstein is responsible for approving major new directions in the investigation, for signing off on the budget and for meeting regularly with Mueller to assess the investigation’s progress.

For more than a year, Trump has railed against the Russia investigation, calling it a “witch hunt” and accusing the prosecutors involved – many of whom, like Rosenstein and Mueller, are Republicans – of harboring ulterior partisan motives.

Trump’s relationship with Rosenstein has been vexed. The president has tweeted that Rosenstein is “weak” but in August he also said the relationship was “fantastic”.

Frustration moved in both directions, with Rosenstein reportedly growing “angry” at Trump in May 2017, after Trump used a memo written by Rosenstein to justify the firing of FBI director James Comey. It was in the sensitive aftermath of that firing that Rosenstein suggested recording the president, the Times reported.

One source told the Times Rosenstein had suggested recording the president “sarcastically”’; others said he was serious.

Sources for the Times report included people familiar with memos kept by former FBI deputy director Andrew McCabe about his conversations with Rosenstein. McCabe was fired by Trump in March, days before he was due to retire.

In a statement on Monday, amid speculation that McCabe desired to undercut Rosenstein – his onetime superior at the justice department who played a direct role in the firing of McCabe’s former boss, Comey – McCabe denied any hand in leaking the contents of his memos to the media.

“To be clear, I had no role in providing information of any kind to the media stories about events following director Comey’s firing,” McCabe said.

The 25th amendment, which was ratified after the assassination of John F Kennedy, provides for various paths of presidential succession, including in case of the president being deemed “unable to discharge the powers and duties of his office” by the vice-president and a majority of cabinet officers.

At a briefing to talk about the UN general assembly, top foreign policy officials were peppered with questions about Rosenstein and the 25th amendment.

“Literally, I have never been in the White House when that conversation has happened,” said the US envoy to the UN, Nikki Haley. “I’m not aware of any cabinet members that are even talking about that. It is completely and totally absurd.”

Secretary of state Mike Pompeo added: “There was no discussion with me about the 25th amendment, so you can now report that there were two senior leaders who said your question is ludicrous.”