Washington (CNN) Deputy Attorney General Rod Rosenstein will meet with President Donald Trump Thursday amid mounting questions about his future and what a dramatic Justice Department shake up would mean for special counsel Robert Mueller's Russia investigation.

Trump, speaking in New York, said Monday he spoke with Rosenstein earlier in the day and anticipated their meeting at the White House once he returned from the United Nations General Assembly.

"I'm meeting with Rod Rosenstein on Thursday when I get back from all of these meetings," Trump said. "And we'll be meeting at the White House, and we'll be determining what's going on. We want to have transparency, we want to have openness and I look forward to meeting with Rod at that time."

Trump's remarks capped off a tumultuous morning in Washington when it was, at times, unclear if the deputy attorney general still had his job.

The confusion began late last week, when Rosenstein and White House chief of staff John Kelly met at the White House in the hours after The New York Times and others reported that he secretly suggested recording the President and discussed recruiting Cabinet members to remove Trump from office.

Rosenstein emphatically denied that he ever sought to record Trump or weighed invoking the 25th Amendment, at first calling the story "inaccurate and factually incorrect." But Kelly relayed to Rosenstein that a firmer denial was required, resulting in a second statement from the deputy attorney general, sources told CNN.

While Trump asked associates in the hours after the story was published if he should "just fire him," the President was also skeptical because the information was reportedly documented in memoranda written by FBI officials, including former FBI Deputy Director Andrew McCabe, whom Trump despises and frequently tells people isn't trustworthy. Although Trump used to remark often that Rosenstein was no ally of his, their relationship had improved in recent months.

Rosenstein's future was a major topic of conversation among senior White House staff over the weekend, but a person close to the President told CNN he was more focused on the controversy surrounding his Supreme Court nominee, Brett Kavanaugh, over the last 48 hours.

Republican allies of Trump Another source said Trump viewed the Rosenstein story with a degree of suspicion, reflecting the suggestion made by Fox News' Sean Hannity and others that it was a "set-up" meant to provoke him to respond. Trump watched Hannity make that argument on Air Force One on Friday evening and asked people about it afterward. Over the weekend,Republican allies of Trump urged him to hold off on a purge of Justice Department officials until after Kavanaugh's confirmation.

Yet Rosenstein overestimated how angry Trump would be after the Times published its reporting, ultimately leading him to offer his resignation to Kelly, according to multiple sources familiar with the matter. Administration officials told CNN that Rosenstein discussed whether he should step down with both Kelly and White House counsel Don McGahn over the last several days.

On Saturday, Rosenstein and Kelly spoke, but the two decided to work out the details on Monday, according to the sources -- explaining that Rosenstein wanted to control the timing and the White House wasn't pressing for it because they were dealing with the Kavanaugh situation over the weekend.

Kelly told associates on Monday that Rosenstein had offered to resign Friday and he had accepted. Yet another source familiar with Rosenstein's thinking told CNN that he expected to be fired by Trump on Monday. Meanwhile, Justice officials began planning for his departure, drafting statements ready to name Solicitor General Noel Francisco acting attorney general for purposes of overseeing Mueller and Matt Whitaker, Attorney General Jeff Sessions' current chief of staff, as the acting deputy attorney general.

Rosenstein -- who had a previously scheduled meeting with senior officials at the White House on Monday -- met with Kelly beforehand to discuss the confusion over his status. At some point while at the White House, Rosenstein and Trump also had an "extended conversation to discuss the recent news stories" at Rosenstein's request, according to a statement later issued by the White House.

One administration official said that part of the calculation fueling Thursday's meeting with the President was a concern on the part of White House officials that only the President, not Kelly, has the authority to fire Rosenstein or accept his resignation.

'Slow-motion Saturday Night Massacre'

The flurry of speculation Monday over Rosenstein's fate prompted a swift reaction on Capitol Hill.

California Rep. Adam Schiff, the top Democrat on the House Intelligence Committee, said on Twitter that Rosenstein should force Trump to fire him rather than resign.

"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," Schiff tweeted.

Under no circumstances should Rod Rosenstein resign. 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. — Adam Schiff (@RepAdamSchiff) September 24, 2018

New York Rep. Jerry Nadler, the top Democrat on the House Judiciary Committee, said to CNN's Kate Bolduan earlier Monday that the developments were "very upsetting" and called it "another step in the unfolding, slow-motion Saturday Night Massacre" -- a reference to a pivotal Nixon-era episode that precipitated former President Richard Nixon's resignation.

One of Rosenstein's top critics in Congress said in a statement as the news unfolded on Monday that the events reinforced GOP calls for access to documents from the Justice Department and FBI.

"Whether or not the latest reports on Rod Rosenstein are true, one thing is clear: what is happening at the Department of Justice is a travesty," said Rep. Mark Meadows, R-North Carolina. "The total lack of transparency and accountability among senior FBI and DOJ officials has devolved into a constant wheel of behind-the-scenes gamesmanship, with anonymous leaks left and right, each seeking to create their own narrative and save face with the public.

Utah GOP Sen. Orrin Hatch told reporters it was up to Trump whether to fire Rosenstein, but said if he did make such a move, "it would cause a furor that I don't think we need right now."

Meanwhile, one of Trump's personal attorneys also weighed in. Jay Sekulow said on his radio show on Monday that if Rosenstein left and a new person was put in place overseeing the Mueller probe, there should be a review and "basically a time out on this inquiry."

'You've got to stand for something'

Trump appointed Rosenstein as deputy attorney general but had expressed extreme frustration with him for months, partly over his decision to hire Mueller last year.

Trump has repeatedly branded the investigation a "witch hunt" and complained that Rosenstein is "conflicted" because he is a witness in the investigation after writing a letter advocating the firing of former FBI Director James Comey over his handling of the Hillary Clinton email investigation.

Rosenstein also signed off on Mueller sending a tax and fraud case against Michael Cohen to the US Attorney's office for the Southern District of New York, a move that ultimately led to an FBI raid on the offices and homes of the President's former lawyer, who is now speaking with prosecutors.

Born in Philadelphia, Rosenstein graduated from the University of Pennsylvania and later Harvard Law School before starting a career at the at the Justice Department spanning nearly three decades.

Rosenstein was confirmed to the deputy attorney general post by an overwhelmingly bipartisan Senate vote of 94-6 last year.

He now oversees the special counsel's investigation into Russia's interference in the 2016 presidential election and any links with the Trump campaign because Sessions is recused from such matters.

Like Sessions, Rosenstein has long has expected that he would leave his post after the midterm elections, according to a source familiar with his thinking.

For Rosenstein, staying put until early November was considered key to give Mueller more time to try to complete his investigation.

With a likely departure on the horizon -- even before the controversy of his comments about wearing a wire and the 25th Amendment became public in news reports -- Rosenstein has been trying to telegraph how he hopes his legacy will be remembered at the Justice Department.

A speech at the department less than two weeks ago struck a somewhat defiant tone in light of the attacks against the Justice Department and the FBI from the President and his congressional allies.

"Most people are familiar with the first clause of our oath, the requirement to 'support and defend the Constitution.' But some overlook the final clause: to 'well and faithfully discharge the duties of the office,'" Rosenstein said.

"The first obligation is generic. It imposes a duty to pursue the national interest over any private interest. That applies equally to all government employees," he said. "But the final clause is specific. Everybody recites the same words, but the meaning varies. In order to well and faithfully discharge the duties of 'the' office, you need to understand the unique responsibilities of your office. What is the mandate of your agency; what is the mission of your component; and how do you add value? You need to know what you stand for. In the words of a classic country song by Aaron Tippin, 'You've got to stand for something, or you'll fall for anything.'"