US President Donald Trump told Russian diplomats that firing the "nut job" Federal Bureau of Investigation director had relieved "great pressure" on him, according to The New York Times.
Mr Trump met with the Russians on May 10, the day after he fired Mr Comey.
The appointment of Mueller as special counsel has drawn generally favorable comments from Democrats and from some Republicans as well.
As for the separate report of a "person of interest" under investigation, the Post said the senior White House adviser "under scrutiny" is someone close to the president but did not name the person.
Rosenstein, who was appointed us attorney by President George W. Bush and held the job for the entire Obama administration, didn't immediately embrace the idea of a special counsel while facing persistent questions from Democratic senators at his confirmation hearing.
In a statement after the Justice Department announcement on Wednesday, Trump said he looked forward to a quick resolution of the matter.
Lawmakers on Capitol Hill welcomed the Justice Department action, but House and Senate Republican leaders said they would go on with their own investigations of the Russian Federation matter.
Rosenstein denied media reports from last week that Comey had asked him for additional resources for his investigation before Trump fired him.
On Thursday, Rosenstein informed the Senate in a closed-door session about Trump's controversial decision to dump Comey.
A third senator, Republican Bill Cassidy, said Rosenstein "indicated what Donald Trump has indicated, that Mr. Trump was leaning in that direction" before Rosenstein presented his memo.
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Rosenstein will give House members a similar briefing Friday morning.
President Donald Trump has lashed out over his Justice Department's appointment of a special counsel to investigate possible ties between his 2016 presidential campaign and Russian Federation, calling it "the single greatest witch hunt" in United States history.
The motivation and timing of Rosenstein's three-page memo has been scrutinized because the White House offered inconsistent explanations for it. But lawmakers at both congressional sessions expressed frustration that Rosenstein would say little in answer to their questions about his actions - or others' - before Comey's firing. Tim Scott, R-S.C. "I do think that the special prosecutor provides a sense of calm and confidence perhaps for the American people, which is incredibly important".
The White House has struggled since Comey's firing to explain the chain of events that led to it and the Justice Department's involvement in that decision. Trump has insisted at times that the decision was his alone, but he also has pointed - as recently as Thursday - to the "very strong" recommendation from Rosenstein.
By several senators' accounts, he contradicted Trump's statements that Rosenstein's written criticism of FBI Director James Comey had been a factor in Comey's recent firing by the president. But Rosenstein added: "I wrote it". "It's going to hamper our ability to get to the bottom of this quickly, I believe".
Rosenstein already made known how he feels about investigators discussing closed cases, in his memo blasting Comey for publicly discussing the closed Clinton email investigation even when it did not yield charges. Rosenstein denounced that as "profoundly wrong and unfair".
House members and senators said Rosenstein steered clear of specifics in answering questions about his appointment of former FBI director as special counsel but made clear that Mueller has wide latitude to pursue the investigation wherever it leads, potentially including criminal charges.
Trump strongly disagreed. The appointment, he said in a briefing with news anchors, "hurts our country terribly".
"The entire thing has been a witch hunt", Trump said Thursday during a White House news conference with Colombian President Juan Manuel Santos.
Asked whether Trump was the victim of a witch hunt, Womack said, "I think there's been an attempt to discredit the president since the day he was elected". Republicans on Capitol Hill hoped the same, reasoning that the appointment of a special counsel could free them to work on a major tax overhaul and other matters without constant distractions.