"Mr. Trump is said to recognize that he needs help beyond the White House counsel".
The White House initially said last week that Trump was prompted to fire Comey after reading the Rosenstein letter.
Other interactions between the White House and Comey illustrate what The Times depicted as an uncomfortable relationship between the two entities.
With a special counsel, Congress loses authority in its role in the investigation.
The President, meanwhile, will announce his selection for Mr. Comey's replacement shortly, with the leading candidate reportedly Joe Lieberman, a former Democrat-turned-independent senator.
But calmness seemed far off.
But Mr. Trump spotted Mr. Comey and called him out.
President Donald Trump has responded with a direct "no, no" to a question about whether he asked former FBI Director James Comey to shelf an investigation into his former national security adviser.
Did he in fact collude with Russian Federation in his campaign to defeat Democrat Hillary Clinton?
He said he did not intend for the document to be a finding of misconduct or a "statement of reasons" to justify the firing, but he added: "I wrote it".
"There is no collusion between, certainly myself and my campaign, but I can only speak for myself and the Russians".
"I respect the move, but the entire thing has been a witch hunt", Trump said.
As special counsel, Mueller will have the freedom and independence to conduct a thorough and complete investigation without the fanfare of partisan politics that has enveloped Washington since January.
Hours after former FBI director Robert Mueller was tapped to take over the investigation, Trump took to Twitter in the early morning to scrub the White House's earlier, more measured, response.
Afterwards, Republican senator Lindsey Graham told reporters that "everything he said was that you need to treat this investigation as if it may be a criminal investigation".
Trump is preparing to leave town Friday on his first foreign trip, and aides have been hopeful the journey will be a chance for the administration to get back on track after weeks of chaos and distractions.
"I think there were lessons learned and they won't be repeated", he said. Trump himself had already contradicted that explanation, telling interviewers earlier that he had already chose to dismiss Comey.
"That's all this is", Trump said, insisting that the Electoral College process is "slanted so much" in Democrats' favor. "I also got a very, very strong recommendation, as you know, from the deputy attorney general, Rod Rosenstein".
"So whether it is the famous Saturday Night Massacre with [Richard] Nixon that led to attention on Watergate, [Ronald] Reagan's Iran-Contra arms-for-hostages scandal, or then with [Bill] Clinton, it was the attempt to obfuscate", he said.
Trott says Rosenstein answered that he didn't want to interfere with the independent investigation that will now be run by former FBI chief Robert Mueller.
With the president departing on Friday afternoon for the first foreign visit of his presidency, focus was turning to the scope and nature of special counsel Robert Mueller's investigation into Russian interference in the USA election and links between the president's election campaign team and the Kremlin.
Wittes also shared Comey's side of a phone call the Federal Bureau of Investigation director had with the President, which Comey described "as weird for how substance-less it was" and included "an undercurrent of Trump's trying to get him to kiss the ring".
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