While a sitting president is unlikely to face criminal prosecution, obstruction of justice could form the basis for impeachment.
His boss, Attorney General Jeff Sessions, recused himself from the Russian Federation investigation earlier this year.
Trump, conservative lawmakers, and his surrogates have made the subject of intelligence leaks their prime focus amid a broad FBI investigation of Russia's interference in the 2016 election.
Trump, however, referred to the Post's reporting as a "phony" story in a tweet Thursday.
"They made up a phony collusion with the Russians story, found zero proof, so now they go for obstruction of justice on the phony story", he wrote on Twitter. "Sad!" Trump tweeted Friday morning.
That includes the finances and business dealings of Jared Kushner, who is married to Trump's daughter Ivanka, and is a senior adviser at the White House, the Washington Post reported, citing unnamed officials familiar with the matter. Journalists across the globe often use anonymous sources to protect confidential sources of information when the information can not be obtained in any other manner.
Trump and the members of his administration have also maintained the publications such as the Post and the New York Times report nothing more than "fake news". Mueller is now leading the investigation into whether Russian Federation interfered in the 2016 presidential election.
Robert Mueller speaks during a farewell ceremony in Mueller's honor at the Department of Justice on August 1, 2013.
On Thursday, Russian President Vladimir Putin said Comey had presented no evidence to prove that Moscow meddled in the United States election, adding that Washington had tried to influence Russian elections "year after year".
Rosenstein could be a witness in the investigation he is overseeing if Mueller has expanded his probe of Russian influence to investigate whether Trump obstructed justice by firing FBI director James Comey.
Roger Stone says President Trump should fire special counsel Robert Mueller and Deputy Attorney General Rod Rosenstein for "wasting the taxpayer's money" in what he called, a "witch hunt" to take down the president.
Trump also appeared to put Deputy Attorney General Rod Rosenstein in his crosshairs. Dianne Feinstein, the panel's top Democrat.
Trump and other officials from his team have long denied that there was any coordination between his campaign and the Kremlin, suggesting that the investigations amount to a "witch hunt".
Did he tell Congress the full story about Mr. Mueller's investigation?
"I think I better wait until I get done with my conversations with Feinstein before I answer that question", Grassley said when asked if the committee would look into obstruction.
A Justice official said Rosenstein was motivated in part because of frustration that recent news stories have unfairly brought on a torrent of "leak" accusations against the Federal Bureau of Investigation and Mueller's team.