Current and former USA officials told the paper that Trump's efforts include confiscating the notes from his interpreter and not allowing them to discuss the details of the meetings with other officials in his administration.
Tillerson had said at the time at a news conference after the two-hour meeting that Trump and Putin covered the war in Syria and other subjects.
If it was not clear already, then a decision by the new chairman of the House Foreign Affairs Committee makes it crystal clear that Democrats will primarily focus on opposing President Donald Trump for the next two years.
"Wow, just learned in the Failing New York Times that the corrupt former leaders of the FBI, nearly all fired or forced to leave the agency for some very bad reasons, opened up an investigation on me, for no reason & with no proof, after I fired Lyin' James Comey, a total sleaze!"
Trump has made a number of decisions during his time as president that have benefitted Putin whilst simultaneously confounding his closest officials and advisors.
According to Trump, "the FBI was in complete turmoil... because of Comey's poor leadership" and the way he handled the investigation into Hillary Clinton's use of a private server to send some government emails.
Trump began the day calling his firing of Comey "a great day for America" in a tweetstorm of outrage over the Times story.
In the same interview with Pirro, Trump referred to the Times' article as "the most insulting article I've ever had written [about me]".
The three investigative committees are apparently not Trump's only worry from a Democratic majority House of Representatives.
Investigators led by former Federal Bureau of Investigation chief Robert Mueller are trying to establish whether there was any collusion between Mr Trump's campaign team and the Russian state to help get him elected.
U.S. Press Secretary Sarah Sanders said in a statement the report is "so outrageously inaccurate it doesn't even warrant a response". She also disputed that Trump had ever been soft on Russian Federation.
"And I can tell you this, if you ask the folks in Russian Federation, I've been tougher on Russian Federation than anybody else, any other ― probably any other president period", he said.
Trump has repeatedly criticized the Mueller probe as a "witch hunt" and views it as an attempt to besmirch the legitimacy of his presidency.
Trump's ex-national security adviser, Michael Flynn, pleaded guilty to lying to investigators about his Moscow ties.
It is unusual for these meetings of these nature to not be reported, as per past presidential standards, the Post notes, especially considering allegations of Russian Federation interfering in the 2016 election.
"Cohen, who will testify in Congress on February 7, insists that he has never been to Prague, but added in a tweet: "#Mueller knows everything!"