Trump has denied the allegations himself, claiming that Cohen is "lying to reduce his jail time" in a tweet where he quoted Fox News correspondent Kevin Corke.
At the same time Cohen was providing the Trumps with news about the business negotiations with Russia, BuzzFeed said, Trump was engaged in his successful presidential election and denying that he had any business deals with Russia.
But Leopold & Cormier are holding on, even trying to pry more information out of the typically-taciturn Mueller with a tweet "urg [ing] the Special Counsel to make clear what he's disputing".
Leading Democrats in the House said they would seek to verify the story, saying that if true it would be evidence of criminal activity by the president.
If Trump were to become only the third USA president in history to be impeached, the Republican-controlled Senate would then decide whether to remove him from office.
Cohen was the president's right-hand man for years at the Trump Organization, the umbrella company, often relied upon to quietly fix hard problems and negotiate deals for the real estate mogul.
Californian Representative Adam Schiff, who is the Chairman of the House Permanent Select Committee on Intelligence released this statement Friday: 'These allegations may prove unfounded, but if true, they would constitute both the subordination of perjury as well as obstruction of justice'.
In court he said his "weakness was a blind loyalty to Donald Trump" - whose "dirty deeds" he had felt compelled to cover up. Harvard Law professor Noah Feldman said Friday in a Bloomberg Opinion piece that ordering Cohen to lie would be "unquestionably the kind of high crimes and misdemeanors contemplated by the Constitution as qualifying for impeachment".More news: Mourinho: Finishing second with Manchester United ‘one of my best achievements’
The story alleges Mr Trump received 10 personal updates from Cohen about a plan to build the Moscow tower at a time when Mr Trump denied having any business ties to Russian Federation.
William Barr, Trump's nominee for attorney general, said at his Senate confirmation hearing Tuesday that a president or anyone else who directs a witness to lie is illegally obstructing an investigation.
Cohen is scheduled to testify to the House Oversight Committee, newly controlled by opposition Democrats, on Feb 7 about his work for Trump.
Democrats, who have taken control of the House of Representatives after the midterm elections, are ramping up investigations of Trump and his administration, while Mueller's probe of whether the Trump campaign colluded with Russian Federation continues quietly behind the scenes.
Earlier Friday Cohen returned to his Manhattan apartment Friday after getting medical treatment - with his arm appearing to be in a sling on a day President Trump and his team once again branded him a liar.
President Donald Trump speaks in the Cabinet Room of the White House in Washington, Monday, April 9, 2018, at the start of a meeting with military leaders.
Much remains unknown about the plans for a Trump Tower in Moscow. Under a Republican-controlled Senate, and needing only a simple majority vote, Barr seems all but certain to be confirmed to the same post he held under President George H.W. Bush. Cohen said the money was paid on Trump's orders.
Democratic Reps. Joaquin Castro of Texas and Eric Swalwell of California also voiced their opinions in tweets, condemning the president's alleged action.