Trump told reporters at the White House that he had "a very good relationship" with Rosenstein and was eager to speak with him aboard Air Force One on a flight to Florida for the International Association of Chiefs of Police conference.
The flight provides an opportunity for their most extensive conversation since news reports last month that Rosenstein had discussed the possibilities in early 2017 of secretly recording Trump to expose chaos in the White House and invoking constitutional provisions to get him removed from office.
Those reports triggered an avalanche of speculation about the future of Rosenstein — and also the special counsel's investigation into possible coordination between Russia and the Trump campaign since the deputy attorney general appointed former FBI Director Robert Mueller to his post and closely oversees his work.
"I didn't know Rod before, but I've gotten to know him," Trump said Monday. "I look forward to flying with him. It'll be very nice."
The Justice Department denied that Rosenstein had proposed invoking the 25th Amendment of the Constitution and issued a statement that said the remark about recording the president was meant sarcastically.
Even so, Rosenstein told White House officials that he was willing to resign and arrived at the White House at one point with the expectation that he would be fired. He met in person with White House chief of staff John Kelly and spoke by phone with Trump during a tumultuous day that ended with him still in his job.
Rosenstein and Trump had been expected to meet at the White House days later, but that meeting was put off so that the president could focus on a confirmation hearing of Supreme Court nominee Brett Kavanaugh. It was not immediately clear if the flight would substitute for the pre-planned White House meeting.
Trump had previously said that he would prefer not to fire the Justice Department's No. 2 official and that Rosenstein has told him he did not say the remarks attributed to him. Advisers had also cautioned Trump against doing anything dramatic in the weeks before the midterm elections next month.
Trump is scheduled to the police chiefs conference in Orlando. The Justice Department in the Trump administration has said reducing violent crime and supporting local enforcement are priorities.
The speculation over Rosenstein's future concerned Democrats, who feared that a dismissal could lead to Trump curtailing Mueller's probe. Although Trump has at times criticized his deputy attorney general, he has reserved his sharpest verbal attacks for Attorney General Jeff Sessions, who recused himself from the Russia investigation in March 2017 because of his own earlier involvement with the Trump campaign.
Both men will likely see their futures re-evaluated after the elections, Trump advisers have said.
Besides the meeting with Trump, Rosenstein has also agreed to a private meeting with House Republicans who want to question him about his reported statements on the president.