If the report is true and if it is confirmed that Trump obstructed justice by trying to get former Federal Bureau of Investigation (FBI) Director James Comey, and perhaps others, to end the Russian Federation probe, that could lead to the President's impeachment by Congress, something that many Democratic lawmakers say they would pursue if circumstances warrant.
It was not immediately clear what prompted the release of that statement by Rosenstein, whether it was in response to recent stories that suggested the President was under investigation for obstruction of justice, or some other reason.
He declined to say whether he thought Mr. Trump sought to obstruct justice, saying that would be up to Mr. Mueller to sort out.
Cullen is a former federal prosecutor who has long ties to former FBI Director James Comey. Last week, Comey testified in a Senate hearing that he believed he was sacked "because of the Russian Federation investigation". Comey said he believes Trump ultimately fired him "because of the Russian Federation investigation".
Mueller's friendship with Comey and Mueller's selection of several investigators who have made campaign donations to Democrats are among the issues Gingrich is raising as problematic.
As recently as Monday morning, the panel on MSNBC's "Morning Joe" ripped Trump for his continuous rants on Twitter, with co-host Mika Brzezinski saying, "He can't help himself". "Witch Hunt", Trump said in a Twitter message.
"Americans should exercise caution before accepting as true any stories attributed to anonymous 'officials, ' particularly when they do not identify the country - let alone the branch or agency of government - with which the alleged sources supposedly are affiliated", Rosenstein said.
Coats met behind closed doors for more than three hours Thursday with the Senate intelligence committee, which is conducting a separate investigation into Russian interference in the 2016 election.
Mr Trump tweeted: "I am being investigated for firing the FBI Director by the man who told me to fire the FBI Director!"
Trump, who hired his own lawyer last month to represent him regarding probes by the special counsel and congressional committees, continued his harsh criticism of the investigations in series of tweets on Friday.
USA intelligence agencies have concluded that Moscow interfered in last year's presidential campaign to try to tilt the vote in Trump's favour.
On Twitter, Mr Trump dismissed as "phony" the idea his campaign colluded with any Russian effort to sway the 2016 USA election.
"The vice president is focused entirely on his duties and promoting the president's agenda and looks forward to a swift conclusion of this matter", Mr Pence's spokesman Jarrod Agen said.