Trump has said he is inclined to let Rosenstein remain in his job for now, though it is expected that he will depart the Justice Department along with Attorney General Jeff Sessions after November's elections.
Trump had also ordered the Justice Department to declassify certain documents Republican lawmakers have requested to be declassified, but changed his mind after the DOJ said it would have a perceived negative impact on the special counsel probe, and "key allies" asked him not to.
The New York Times reported last Friday that Rosenstein had suggested secretly recording Trump to document the chaos of the White House in 2017 and had raised the issue of removing the president from office. "It undermines Rosenstein's credibility and Trump's ability to trust his closest advisers".
No meeting with Rosenstein appeared on the president's schedule, which was released by the White House.
Initially, Trump had said that a face-to-face meeting would take place Thursday, after he returned to the White House from three days in NY for meetings at the U.N. General Assembly. Mark Meadows of North Carolina and Jim Jordan of OH, said they believed Rosenstein would be coming for a transcribed interview as part of the congressional investigation into the FBI's handling of the Hillary Clinton and Russian Federation investigations.
The White House meeting between the president and deputy attorney general was originally scheduled in the wake of a flurry of conflicting reports Monday saying Rosenstein had either quit or was set to be fired.
"They do not want to do anything to interfere with the hearing" taking place on Thursday on Capitol regarding sexual assault allegations by Christine Blasey Ford against Trump's Supreme Court nominee, Brett Kavanaugh.
Rosenstein has denied the story as "inaccurate and factually incorrect".
The president first brought up the possibility of moving his meeting with Rosenstein at a marathon press conference on Wednesday.
"Not wanting to fire Rod Rosenstein is consistent with what I have understood for weeks, not just days", said Rep. Mark Meadows, a North Carolina Republican who talks to Trump often.
Over the weekend, Rosenstein resigned verbally to White House Chief of Staff John Kelly, who accepted the offer, but after the phone call with Trump on Monday, Rosenstein made a decision to stay on, people familiar with the matter have said.
Even if Rosenstein survives the week, it's not clear how much longer he'll be around. Rosenstein isn't likely to agree to such a demand, increasing the risk that he'd resign, the person said.