Rosenstein said he is the individual who now has the authority to hire or fire Mueller after Attorney General Jeff Sessions recused himself from any investigation into Russia amid questions about his contacts with Russian officials during the President Donald Trump's campaign. "The investigation has been cloaked in secrecy and it's unclear how many others have been questioned by the FBI".
After Mr Comey's testimony, Mr Trump said he had been vindicated because his former FBI director confirmed telling Mr Trump on three occasions that he was not under investigation.
"I think he's considering perhaps terminating the special counsel", Chris Ruddy told PBS's Judy Woodruff on "PBS NewsHour".
The Post cited five officials who it says spoke to it on condition of anonymity.
Sessions' testimony came in the wake of fired FBI Director Comey's assertion that he believed he was sacked by Trump as part of an effort to influence the agencies' investigation into Russia's meddling in the presidential polls.
The committee leaders said in a statement that they "look forward to future engagements" with Mr Mueller.
According to officials, investigators are also checking to see if there is any evidence that Trump associates committed any financial crimes.
Former FBI director James Comey told Congress last week he believed he was sacked by Trump to undermine the agency's Russian Federation probe.
Sessions' appearance before the intelligence committee is an indication of just how much the Russian Federation investigation has shaded his tenure.
"I have never been directed to do anything I believe to be illegal, immoral, unethical or inappropriate", Rogers said.
A spokesman for the office of the special counsel declined comment.
Under Justice Department regulations created to ensure a measure of independence for the special counsel, Rosenstein may only fire Mueller for "good cause". The President has stressed he was not under investigation, and has taken to Twitter to dismiss the entire investigation as a hoax.
According to the Post, it did change after Trump fired Comey.
At a hearing last week, both Coats and Rogers said they were not pressured by the Trump administration.