Mr Trump - who has repeatedly denied any collusion with Russian Federation - said the move was the latest action in a "phony story".
Robert Mueller, the special counsel probing Russia's interference in the 2016 United States presidential elections and its possible collusion with figures in the Donald Trump campaign, is reportedly investigating the financial dealings of the president's senior adviser and son-in-law, Jared Kushner.
Comey, who had been overseeing the bureau's Russian Federation investigation, says he had previously assured Trump he was not personally under investigation.
If true, the investigation represents an expansion of Mr Mueller's original probe into links between Moscow and the 2016 Trump presidential election campaign. Last week, former FBI Director James Comey told Congress he believed he was sacked by Trump to undermine the agency's Russian Federation investigation.
Moscow has denied any interference and the White House has denied any collusion.
Democrats note that after the Justice Department appointed a special prosecutor to investigate the Watergate break-in during the in the 1970s, then-President Richard Nixon ordered the department to fire the special prosecutor, Archibald Cox, prompting his attorney general and deputy attorney general both to resign in what became known as the "Saturday Night Massacre".
Trump Confirms He's Under Investigation Over Comey Firing
And it's up to him to decide how much to tell Rosenstein about the substance of the investigation itself. Trump has expressed increasing dissatisfaction with the Justice Department since Mueller's appointment.
According to the Post, it did change after Trump fired Comey.
Kushner has been given a wide range of responsibilities in the Trump White House, from restructuring the federal government to achieving Middle East peace.
Law enforcement sources tell CNN that the special counsel is gathering information and considering whether there is evidence to launch a full-scale obstruction investigation.
"We do not know what this report refers to", Kushner's attorney Jamie Gorelick told The Post in an email.
Separately, the House committee said former Homeland Security Secretary Jeh Johnson would appear at a June 21 public hearing on the Russian Federation probe.
"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.
Peter Carr, a spokesman for Mueller, did not comment on the Post report when reached by Fox News, but did say that the special counsel's office "has undertaken stringent controls to prohibit unauthorized disclosures that deal severely with any member who engages in this conduct".