In another early morning tweet, Trump said that despite months of investigation no proof has been presented of any collusion he may have had with the Kremlin.
Comey was sacked by the president on May 9 while he was overseeing the investigation into potential Russian interference into last year's election and potential collusion with the Trump presidential campaign.
Trump, however, referred to the Post's reporting as a "phony" story in a tweet Thursday. Trump appeared to take a subtle swing at Rosenstein during his tweets today, and new reports are saying that the Deputy Attorney General might recuse himself from the Russian Federation probes as well.
Mueller serves at the will and pleasure of Deputy Attorney General Rod Rosenstein, who appointed him to head the inquiry.
The White House confirmed Mr Mueller was among the candidates the president interviewed to replace Mr Comey.
Coats told associates that Trump asked him if he could intervene with Comey to get the Federal Bureau of Investigation to back off its focus on former national security adviser Michael Flynn, according to officials.
On Thursday, Vice President Mike Pence hired an outside lawyer to advise him and represent his legal interests. Earlier this week the Washington Post released a story reporting Mueller is investigating President Trump for obstruction of justice.
Trump's tweet Friday morning came after a series of angry tweets Wednesday in which he also complained about a "witch hunt" and criticized the investigation into ties between his campaign and Russia as "a phony collusion with the Russians story".
Those who have agreed to be interviewed include: Dan Coats, director of national intelligence; Mike Rogers, chief of the National Security Agency; and Richard Ledgett, who recently left his post as deputy to Rogers.
In another tweet this week, Mr. Trump had said: "After 7 months of investigations & committee hearings about my 'collusion with the Russians, ' nobody has been able to show any proof".
He was answering accounts that he is personally being investigated for obstruction of justice with a characteristic scorched-earth defense claiming mistreatment of historic proportions and calling into question the probity of his accusers.
Trump himself said after the firing that he had made up his mind to fire Comey regardless of Sessions' and Rosenstein's arguments.
"Crooked H destroyed phones w/ hammer, "bleached" emails, & had husband meet w/AG days before she was cleared- & they talk about obstruction?" the president said in a follow up tweet. "Nice", Trump wrote in his first tweet.