Mr Rosenstein was reported last week to have suggested secretly recording Mr Trump
Rod Rosenstein to meet Donald Trump as job hangs in balance
US Deputy Attorney General, Rod Rosenstein, who oversees the special counsel investigation into Russia meddling in the 2016 election, will meet President Donald Trump on Thursday amid reports that he is close to losing his job.
The pair will meet to discuss reports that he suggested taping the president or that he was unfit for office.
"We'll be meeting at the White House and we'll be determining what's going on," Mr Trump said at the UN General Assembly. "We want to have transparency, we want to have openness, and I look forward to meeting with Rod at that time."
Any termination or resignation would have immediate implications for special counsel Robert Mueller's investigation of possible collaboration between Russia and the Trump campaign before the 2016 election. Mr Rosenstein appointed Mueller and oversees his investigation.
Mr Rosenstein reportedly spent the weekend contemplating his resignation following the revelations, and reports of his imminent dismissal from the administration emerged on Monday.
The two spoke on the phone on Monday, White House Press Secretary Sarah Sanders said.
"At the request of Deputy Attorney General Rod Rosenstein, he and President Trump had an extended conversation to discuss the recent news stories." Mrs Sanders said in a statement posted on Twitter.
"Because the President is at the United Nations General Assembly and has a full schedule with leaders from around the world, they will meet on Thursday when the President returns to Washington DC."
It is unclear whether Mr Trump would accept his departure given that it would plunge his administration into uncertainty ahead of the crucial midterm elections in November, although the two men have not always agreed.
Questions over Mr Rosenstein's position in the Trump administration come after he was reported to have suggested he should secretly record Mr Trump, and invoke the 25th Amendment, which allows a sitting president to be removed from office is he is deemed to be unfit.
The report by The New York Times said Mr Rosenstein made the suggestions in 2017 following the firing of James Comey as FBI director, in which a memo by Mr Rosenstein was used as justification, to his annoyance. The Deputy Attorney General said he made the comment sarcastically.
US media outlets reported different versions of events, some saying Mr Rosenstein had given his resignation, with others saying he insisted the White House would have to fire him.
White House officials told the Axios news site that Mr Rosenstein verbally gave his resignation, although that report was later disputed.
Mr Rosenstein took on the role as the Department of Justice's top official overseeing Robert Muller's probe into Russian interference in the 2016 election when Attorney General Jeff Sessions recused himself from the investigation after his own connections with the Russian government while working on the Trump campaign became public. He is the second most senior official in the Department of Justice.
Mr Rosenstein has often suffered the ire of Mr Trump as a defender of Mr Muller's investigation, refusing to consider firing the special counsel, despite the president's claims it is a politically-motivated "witch hunt".
Mr Trump says Mr Rosenstein is "conflicted" in his dealing with the probe because he is a witness in the investigation after writing a memo arguing for the firing of FBI director James Comey.
Former FBI Deputy Director Andrew McCabe said he was "deeply concerned" about the reports that Mr Rosenstein was stepping down, indicating his departure would put Mr Muller's probe at risk.
"There is nothing more important to the integrity of law enforcement and the rule of law than protecting the investigation of Special Counsel (Robert) Mueller," Mr McCabe said in a statement. "If the rumours of Deputy AG's Rosenstein's departure are true, I am deeply concerned that it puts that investigation at risk."
If Mr Rosenstein resigns, Mr Trump has more leeway on replacing him while firing him would make it harder for Mr Trump to designate a successor.
Noel Francisco, the solicitor general, would assume responsibility for overseeing Mr Muller's probe if Mr Rosenstein leaves the administration. Matthew G. Whitaker, the chief of staff to Attorney General Jeff Sessions would take on the roles of Deputy Attorney General.
The tumult over Mr Rosenstein's status will be an unwelcome domestic distraction for Mr Trump as he turns his attention to foreign affairs ahead of the United Nations General Assembly in New York.
Mr Trump will focus on the Middle East during his time at the General Assembly. Several sideline meetings with heads of state are planned including with Israel and Egypt. Mr Trump will also chair a meeting of the Security Council on nuclear proliferation, which is expected to focus primarily on Iran.