Deputy Attorney General Rod Rosenstein, speaking before lawmakers Friday, defended his controversial memo criticizing former FBI Director James Comey - a document the White House initially used to justify the firing of Comey.
White House Press Secretary Sean Spicer, who was on Air Force One with Trump when both stories broke, issued written statements responding to them. "He was insane, a real nut job", Trump told Russian Minister Sergei Lavrov and the country's Ambassador to the USA, the Times reported, citing minutes of the meeting between Trump and the Russians.
And The New York Times said the USA president had told top Russian officials Mr Comey's sacking had relieved "great pressure" on him.
The appointment of Mueller as special counsel has drawn generally favorable comments from Democrats and from some Republicans as well.
It did not deny the Times report that Trump was critical of Comey to the Russians the day after he fired him. The President's decision was widely viewed as an attempt to undercut the FBI's probe into whether his campaign team colluded with Russian Federation to rig the 2016 presidential election in his favor, as well as subsequent reports that he demanded Comey pledge loyalty and pushed him to drop an investigation of his former national security adviser have only strengthened that view, which led to accusations that he's obstructed justice - a potentially impeachable offense.
FBI agents assigned to the investigation are said to remain "keenly interested" in Trump aides who were influential during last year's political campaign but either did not join the new administration or have already left the government, such as former campaign chairman Paul Manafort and former national security adviser Michael Flynn.
However, the story says, "people familiar with the investigation said the intensifying effort does not mean criminal charges are near, or that any such charges will result".
The FBI's investigation has bedeviled the Trump administration, and the president personally.
Rosenstein ensured House members that the Justice Department would work with members of Congress "to coordinate our parallel efforts" and establish a point of contact for Mueller.
In the days after Comey's firing, some lawmakers, former Justice officials and people who knew Rosenstein wondered if he had been forced to write the politically charged memo for Trump.
"There was considerable frustration in the room", said Rep. Seth Moulton, D-Mass., an Iraq war veteran and member of the Armed Services Committee.
The White House has struggled to explain the chain of events that led to Mr Comey's firing and exactly who made the decision.
Rosenstein made it abundantly clear to the lawmakers that he drafted his memo only after Trump told him of his plans to dismiss the Federal Bureau of Investigation director. Trump has insisted at times that the decision was his alone, but he also has pointed to the "very strong" recommendation from Rosenstein. "I believe it. I stand by it".
Trump added, "I'm not under investigation". Rosenstein denounced that as "profoundly wrong and unfair". Trump hopes to use his appearances in the Mideast to define the conflict with Islamic State and other terrorists as one of good versus evil, rather than the West versus Islam, the official said. However, at a combative news conference Thursday, he fell short in trying to resolve questions about investigations into his campaign and his first four months in office.