Appearing on CNN's New Day, Samantha Vinograd said that it was puzzling - or maybe not - why Trump would be hiding what transpired between the two leaders when they met in private with only interpreters on hand.
In a strongly worded reaction, White House press secretary Sarah Sanders said, "This is absurd".
Later, at the same G-20 summit, Trump and Putin spoke at a banquet with only Putin's interpreter present.
He said he had talked in Helsinki to the Russian leader "like every president does" and that they had had a "great" conversation about "very positive things" such as the economy.
Trump also took the opportunity during the 25-minute interview to touch on several other topics including the government shutdown, the US-Mexico border, and a potential race against former Vice President Joe Biden.
"If the president knowingly puts himself in the room with Vladimir Putin again, fails to read out his team again on what happened, you have to imagine that Bob Mueller is going to be looking into the intent and motivations behind those actions", she speculated.
The New York Times report Friday cited unnamed former law enforcement officials and others familiar with the investigation.
On Friday, the New York Times published a story about an FBI probe into Trump's behavior following his firing of former FBI Director James Comey.
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.
CNN reported last month that the dismissal set off a flurry of activity by Rosenstein and top Federal Bureau of Investigation officials. "I think it's the most insulting thing I've ever been asked", he said, without directly answering the question. But Trump's behavior in the days around Comey's May 2017 firing as FBI director, specifically two instances in which he seemed to tie Comey's ousting to the Russian Federation investigation, helped trigger the counterintelligence part of the investigation, according to the newspaper. "Unlike President Obama, who let Russian Federation and other foreign adversaries push America around, President Trump has actually been tough on Russian Federation". Trump has repeatedly slammed the Mueller investigation as a "witch hunt" and insisted there was "NO COLLUSION" with Russian Federation.
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 advisor, Michael Flynn, has pleaded guilty to lying to investigators about his Moscow ties.
Trump's former presidential campaign chairman, Paul Manafort, has been convicted in one case brought by Mueller and pleaded guilty in another, over financial crimes related to his work in Ukraine before the 2016 campaign, and for witness tampering.
Cohen will testify publicly before a House committee next month.