WASHINGTON—President Trump told advisers he is open to keeping Deputy Attorney General Rod Rosenstein on the job, and allies of the No. 2 Justice Department official said Tuesday he has given them the impression he doesn’t plan to quit.
That raised at least the possibility that a roller coaster of a week could end with no major shake-up in the top ranks of the Justice Department, even as White House and Justice officials cautioned that it was impossible to know for sure what Mr. Trump would do. Mr. Rosenstein oversees special counsel Robert Mueller’s investigation into Russian meddling in the 2016 election and any links to Trump campaign officials. Mr. Trump has dismissed the probe as “a witch hunt.”
The president has told advisers that he wants to hear directly from Mr. Rosenstein about reports that he discussed secretly recording the president and recruiting cabinet members to remove him from office, according to people who have spoken to the president. That meeting is scheduled for Thursday afternoon.
The president’s willingness to hear out Mr. Rosenstein signaled to advisers that he harbors doubts about whether the top official in fact sought to have him ousted him from the Oval Office, these people said. The issue arose after the New York Times reported that Mr. Rosenstein floated the idea in early 2017, something he has strongly denied.
Republicans in the House were preparing a subpoena for memos allegedly detailing Mr. Rosenstein’s comments on surreptitiously recording the president, according to a person familiar with the matter.
Mr. Rosenstein has been a target of the president’s ire as part of his disdain for Mr. Mueller’s investigation, but those tensions eased in recent months, White House officials said. According to people familiar with the matter, aides have counseled the president that Mr. Rosenstein is cut from a different cloth than James Comey or Andrew McCabe, two former FBI officials who have been sharply criticized by Mr. Trump.
“The president is genuinely conflicted,” said one person who has spoken to the president. “He’s got an open mind about whether Rod really tried to orchestrate this.”
The should-he-stay-or-should-he-go speculation over Mr. Rosenstein has gripped Washington because of the implications of ousting the man overseeing the Mueller investigation probe.
White House officials said Thursday’s meeting between the two men could focus on terms of a resignation, which they said Mr. Rosenstein offered to White House chief of staff John Kelly last week. During a meeting with Mr. Kelly on Monday, Mr. Rosenstein said he wanted to speak with the president about why he would step down and discuss how his exit would be publicly described, according to people familiar with the matter.
People close to Mr. Rosenstein took issue with this depiction of Thursday’s agenda, however. And as Mr. Rosenstein talked to White House officials over the weekend about his possible departure, he spoke to friends who came away with the impression he had no immediate plans to step down.
White House officials said there was a real possibility that the president would decide to keep Mr. Rosenstein in his job after the meeting. “We’re ready for any and all possibilities,” one official said.
The Justice Department declined to comment Tuesday on what Mr. Rosenstein planned to tell Mr. Trump at the coming meeting.
Rudy Giuliani, a lawyer for Mr. Trump, said regardless of whether Mr. Rosenstein stays in his job, the Mueller probe should be paused and examined “in the interest of fairness.”
“I’m not sure they should get rid of him,” Mr. Giuliani said of Mr. Rosenstein. “But I do think they should take a serious look at whether he should be the decision maker.” Among other factors, he said, Mr. Rosenstein was a witness in the investigation, given his role in the president’s May 2017 decision to fire Mr. Comey as FBI director.
On Capitol Hill, some Republican lawmakers are urging Mr. Trump not to fire Mr. Rosenstein, an action they fear would ignite a political storm six weeks before the congressional elections. Mr. Trump is aware of concerns among Republicans, inside and outside the White House, that Mr. Rosenstein’s departure would give rise to suspicions that Mr. Trump was trying to derail the Russia investigation, said one person familiar with the matter.
“If there’s any attempt to fire or force out Deputy Attorney General Rosenstein, that would be a huge red flag and very problematic,” said Sen. Susan Collins (R., Maine).
Sen. John Cornyn of Texas, the No. 2 Republican in the Senate, said he was “glad the president and Rosenstein are talking.” He added, “I hope they work it out.”
Sen. John Kennedy (R., La.), an ally of Mr. Trump’s on Capitol Hill, told reporters he may phone the White House to tell the president he believes Mr. Rosenstein’s denials.
“It’s just another sideshow in the circus,” Mr. Kennedy said.
—Joshua Jamerson and Aruna Viswanatha contributed to this article.