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If Trump fires Rosenstein — as he has reportedly considered doing multiple times in recent months — the position would normally be filled by the Department of Justice's No. 3 official, the associate attorney general. But the last person to hold the job, Rachel Brand, resigned in February, and a permanent replacement has yet to be appointed.
If Rosenstein was fired under such circumstances, Solicitor General Noel Francisco would take his place.
Francisco could then be thrust into the national spotlight and tasked with one of the most politically charged responsibilities in modern history: firing the special counsel at the president's request.
Despite his high standing, Francisco has kept a low profile at the department. Some experts have suggested that Francisco could be more amenable to Trump's whims than Rosenstein, who appointed Mueller in the first place in May 2017.
Trump has recently accused both Mueller and Rosenstein of having conflicts of interest that preclude them from fairly completing the investigation.
Francisco, on the other hand, recently intervened in a Securities and Exchange Commission case and reasserted Trump's constitutional ability to hire and fire basically all federal authorities, the Los Angeles Times reported Sunday.
"The Constitution gives the president what the framers saw as the traditional means of ensuring accountability: the power to oversee executive officers through removal," Francisco said in the case, the newspaper reported.
However, Francisco also joined Rosenstein and Attorney General Jeff Sessions for a dinner in February that was widely viewed as a symbol of defiance against Trump. A decision by Sessions, a favorite target of Trump's anger, had been called "disgraceful" by the president in a tweet earlier that day.
If Francisco is for any reason unable to helm the role of deputy attorney general, the line of succession will proceed as follows:
A recent wave of administration attrition, coupled with Trump's public broadsides against the Justice Department and a growing consensus that the way to fire Mueller is through Rosenstein, have all fueled speculation that the No. 2 official is under threat.
The White House said last month that there have been "no discussions or consideration" about firing Rosenstein. But White House spokeswoman Kellyanne Conway refused to say Rosenstein's job was safe in a CNN interview Monday morning.
The White House did not immediately respond to CNBC's request for comment.
Rosenstein drafted a private three-page memo to the president criticizing former FBI Director over his handling of an investigation into former presidential candidate Hillary Clinton's email server.
The memo was ultimately used as the basis for the White House to justify firing Comey — but Rosenstein pushed back at the time, saying, "My memorandum is not a statement of reasons to justify a for-cause termination." He has since clashed with Trump by reiterating his confidence in Mueller as leader of the investigation into potential collusion between Trump's presidential campaign and the Kremlin.
At a White House briefing last week, press secretary said Trump "certainly believes" he has the power to fire Mueller himself. But the Code of Federal Regulations says "the Special Counsel may be disciplined or removed from office only by the personal action of the Attorney General."
Sessions, who would be in charge of the special counsel under normal circumstances, ceded responsibility to Rosenstein when he recused himself from the probe after failing to disclose in congressional testimony his contacts with a Russian ambassador.
Even if a new official were to assume control of the special counsel investigation, it's far from guaranteed that firing Mueller would halt the probe.
"So much has already happened," said Chris Edelson, an assistant professor of government at American University. Mueller's investigation, for instance, has already delivered more than a dozen indictments and collected five guilty pleas as part of the probe; if the special counsel is dissolved, Edelson said it's not clear what would become of those cases.
"Even if you don't shut down some or all federal investigation, state or local operations could continue," Edelson said.
He referenced the investigation into Trump's longtime personal lawyer, Michael Cohen, whose office and residence were raided by FBI agents last week. Lawyers for both Cohen and Trump have argued that many of the seized documents are protected by attorney-client privilege. Cohen is scheduled to attend a U.S. District Court hearing in New York on Monday for the first time, after failing to appear at the first hearing on Friday.
The raids were the result of a search warrant initially referred by Mueller, and signed off on by Rosenstein.
"It's worked for him in the past to sort of challenge norms and challenge the rule of law," Edelson said, pondering whether Trump could simply forego the rules and ax the investigations outright.
"I don't mean to be paranoid about this, but I'm assuming that to some extent rules will apply," he said.
— NBC News' Pete Williams contributed to this report.