Francisco, a former clerk to Supreme Court Justice Antonin Scalia, has argued against appointing special counsels to investigate White House scandals.
Mr. Trump blames Sessions for recusing himself in the Russian Federation investigation, which is why Rosenstein is in charge of the probe.
"Making Trump fire him would be a good way to get a parting shot by Rosenstein", Blake adds.
Rosenstein was captured by photographers leaving the White House after his meetings Monday and was led out by Kelly.
Deputy Attorney General Rod Rosenstein expects to be fired following the reports about his early tenure within the Justice Department but had not actually stepped down as he traveled to the White House on Monday morning for a meeting.
Mr Trump himself was in NY for a meeting of the UN General Assembly.
For more than a year, Trump's public and private comments about the Russian Federation probe have led to speculation and concern that Rosenstein could be fired.
After being accused by Jordan of "hiding" information, Rosenstein wagged his finger at the congressman and said his department was "working around the clock" and only interested in truth.
Nevertheless, if you're concerned about protecting the independence of the investigation-or would just prefer to see the process play out without interference-there are a few things you can do in the event that Rosenstein is ousted. A spokesman for Mueller declined to comment.
Rosenstein says his comments are being incorrectly reported and that anything he said was meant to be sarcastic.
The allegations, depicting a panic-stricken No. 2 official at the Justice Department who has been a target of the president in the past, immediately raised questions about Rosenstein's future and how Trump would react.
"The New York Times' story is inaccurate and factually incorrect", Rosenstein said. That report and an unsigned opinion piece by a senior official in the Republican administration played to some of the president's concerns about a secret "Deep State" trying to undermine him from within the government.
The administration official, whom Trump has called for a federal investigation to unmask, wrote that there was a group of officials working to safeguard the country from the president's most risky impulses.
The 25th Amendment provides for the removal of a sitting president if he or she is judged unfit to carry out the duties of the office.
"Based on my personal dealings with the president, there is no basis to invoke the 25th Amendment", Rosenstein said.
Others involved in those May 2017 discussions said Rosenstein's comments about secretly recording the president were sarcastic, and came as McCabe was pressing the Justice Department to open an investigation into the president over the firing of Comey as possible obstruction of justice.
If Rosenstein were to resign instead of being fired, it would allow the White House to use the Vacancies Reform Act to fill his role.
"This started with the firing of Comey, the firing of McCabe, the incessant attacks on Attorney General Sessions, the incessant attacks on the Federal Bureau of Investigation, on people like Bruce Ohr, on others, the incessant attacks on the institutions that we depend on to do our police work and protect the integrity of our system", Nadler told CNN's Kate Bolduan as news of Rosenstein's uncertain future developed.