Comey told a Senate panel last week he believed Trump fired him because of the Federal Bureau of Investigation's Russian Federation probe. Comey testified that after security adviser Flynn was forced to resign, at one meeting with the president alone inside the Oval Office on February 14, Trump said to him, "I hope you can let this go".
The interest in Kushner's activities by the special counsel's office is no more extensive than the FBI's interest in his multiple roles on both the Trump campaign and transition team, which CNN reported last month. The report came quietly but abruptly from Special Counsel Robert Mueller, who was brought to Capitol Hill shortly after Comey was sacked in order to fill in for Attorney General Jeff Sessions.
According to Comey, Trump was referring to the FBI's investigation into Mike Flynn, who was sacked himself as national security adviser months earlier for allegedly lying to White House officials about his post-election contacts with Russian operatives. Several congressional committees are also investigating the Russian Federation matter.
Citing unnamed sources, ABC reported Rosenstein told Associate Attorney General Rachel Brand she would have authority over the probe if he were to step aside.
Mr Mueller was appointed as special counsel to investigate Moscow's involvement in the presidential election past year and possible collusion by Trump campaign associates with the Russians, but there has been no indication that Mr Mueller told the president to fire Mr Comey.
A Trump confidant said this week the president had considered firing Mueller.
"Why is that Hillary Clintons family and Dems dealings with Russian Federation are not looked at, but my non-dealings are?" he asked at one point.
"The Rosenstein statement suggests there must be a heck of a Trump story coming based on alleged information from anonymous foreign officials", tweeted conservative commentator Bill Kristol.
Moscow has denied any interference.
The White House has said that the subsequent meeting with the banker, Gorkov, was a pre-inauguration diplomatic encounter unrelated to any business matters.
"After 7 months of investigations & committee hearings about my "collusion with the Russians", nobody has been able to show any proof".
Before he met with the US President, Comey gathered his leadership team at the FBI and discussed the assurances that would be given to Trump regarding investigations against him.
The Post reported that FBI agents and federal prosecutors are also looking into the financial dealings of Flynn, former campaign chairman Paul Manafort and former campaign policy adviser Carter Page.
Later, that position was contradicted by the President himself, in an interview with NBC's Lester Holt, when he stated that he had made up his mind to fire Comey before he had even seen the Rosenstein memo.
Vice President Mike Pence has also hired a private lawyer to represent his interests in the expanding probe.
President Trump on Friday continued to belittle the investigations of his White House, again using the term "witch hunt" to describe them.
The transition official said the organization has also separately asked the General Services Administration to preserve records from the Trump transition that were transferred to its facilities after the inauguration.