The Post writes that Trump had on at least one occasion taken the notes of his own interpreter and told them not to mention what had been talked about with Putin, according to current and former US officials.
According to the Post there is no detailed record of Trump's personal talks with Putin at five locations over the past two years.
"I call him "Lying James Comey" - he did a bad job as Federal Bureau of Investigation director", Trump told Pirro.
US President Donald Trump has denied hiding details of conversations with his Russia's Vladimir Putin, saying he is "not keeping anything under wraps". I'm not keeping anything under wraps. "These people make it up", Trump said.
"Well, I would (release details of the conversations with Putin)..."
After a meeting with Mr Putin in Hamburg in 2017, Mr Trump took possession of his interpreter's notes, the paper says.
But Trump's behaviour in the days around Comey's May 2017 firing as FBI director, specifically two instances in which he seemed to tie Comey's removal to the Russian Federation investigation, helped trigger the counterintelligence part of the investigation, according to the newspaper.
Trump faced widespread criticism following the Helsinki summit from both Democrats and Republicans, with his performance at a press conference alongside Putin - in which he sided with Moscow over his own government's assertion that the Kremlin meddled in the 2016 U.S. election, and heaped praise on the Russian Federation strongman - roundly condemned by allies and critics alike.More news: Meghan Markle's Half-Brother Arrested for DUI
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Trump also took aim at an earlier story in The New York Times stating that the FBI launched a previously undisclosed counterintelligence investigation to determine whether he posed a national security threat, at the same time that it opened a criminal probe into possible obstruction of justice.
"James Comey was sacked because he's a disgraced partisan hack, and his Deputy Andrew McCabe, who was in charge at the time, is a known liar fired by the FBI", Sanders said.
A White House spokesman disputed that characterisation and said that the Trump administration has sought to "improve the relationship with Russia" after the Obama administration "pursued a flawed "reset" policy that sought engagement for the sake of engagement".
"Over time it got harder and harder, I think, because of a sense from Trump himself that the leaks of the call transcripts were harmful to him", said a former administration official.
Trump said that it was "the most insulting article I've ever had written [about me]" on Saturday, and asserted that he has been "tougher on Russian Federation than anybody else, any other - probably any other president period". "And if you read the article, you'd see that they found absolutely nothing".
"The Washington Post is nearly as bad, or probably as bad, as the New York Times", Trump said.
"I mean it's so ridiculous", he added, before taking a swipe at Washington Post owner Jeff Bezos, who has been in the news lately for an ugly divorce after his sexts with former "Extra" and Fox News host Lauren Sanchez surfaced.
Trump has repeatedly slammed the Mueller investigation as a "witch hunt" and insisted there was "NO COLLUSION" with Russian Federation.