(CNN) Robert Mueller's appointment as special counsel to lead the Russia probe in May caught President Donald Trump by surprise. Seven months later, the President's defenders have gone into overdrive hoping to discredit the investigation as Trump insists publicly he has no plans to fire Mueller.

Trump has called the investigation a "witch hunt," his allies on Capitol Hill highlight the political contributions Mueller's team members have made to Democrats over the years, and Fox News banners muse about an anti-Trump "coup in America?" Trump transition lawyers also say Mueller's team wrongfully got a hold of tens of thousands of emails

Trump, his lawyers, his Cabinet, and White House staff all still maintain that Mueller isn't on the chopping block even as his investigation reaches members of the President's inner circle.

But what's to stop the President from handing Mueller a pink slip if he changes his mind?

Can Trump fire Mueller?

The answer isn't straightforward.

Under the special counsel regulations, Mueller may be "disciplined or removed from office only by the personal action of the attorney general."

Attorney General Jeff Sessions has recused himself from all matters related to the 2016 presidential campaigns, so the power to fire Mueller falls to Deputy Attorney General Rod Rosenstein.

But Rosenstein has continuously offered a full-throated defense of Mueller's integrity, telling the House Judiciary Committee just last week that he's seen no "good cause to fire Mueller."

Trump does have the ability to fire Sessions and/or Rosenstein -- as a members of the executive branch -- in which case Associate Attorney General Rachel Brand would elevate to acting attorney general.

"The President is not considering changes to the Department of Justice leadership," Raj Shah, principal deputy White House press secretary, told The Washington Post Monday.

But former acting solicitor general Neal Katyal, who helped draft the special counsel regulations in the 1990s, has also suggested that the rules don't "foreclose the possibility of political interference in the investigation."

"Our Constitution gives the President the full prosecution power in Article II; accordingly, any federal prosecutor works ultimately for the President," Katyal said. "The President, therefore, would have to direct Rosenstein to fire Mueller -- or, somewhat more extravagantly, Trump could order the special-counsel regulations repealed and then fire Mueller himself."

When President Richard Nixon ordered Attorney General Elliot Richardson to fire Archibald Cox, the special prosecutor investigating Watergate, it proved to be a turning point in the investigation, leading to the infamous "Saturday Night Massacre."

Richardson refused and resigned in protest, leading Nixon to order then-Deputy Attorney General William Ruckelshaus to fire Cox. But Ruckelshaus also refused and resigned, and eventually then-solicitor general Robert Bork fired Cox.

JUST WATCHED Pro-Trump media's war on Mueller Replay More Videos ... MUST WATCH Pro-Trump media's war on Mueller 05:36

Is Trump worse off if he tries?

Harvard Law School Professor Jack Goldsmith, former head of the Justice Department's Office of Legal Counsel during George W. Bush's administration, has suggested the President will only come under further scrutiny if he tries to fire Mueller.

"I don't see how firing Mueller gives Trump relief from the investigation. More likely the opposite, since it would call Trump into greater suspicion. Just as it got worse for him after he fired (former FBI Director James) Comey, it would get yet worse for him if he fired Mueller," Goldsmith tweeted.

6/ So I don't see how firing Mueller gives Trump relief from the investigation. More likely the opposite, since it would call Trump into greater suspicion. Just as it got worse for him after he fired Comey, it would get yet worse for him if he fired Mueller. — Jack Goldsmith (@jacklgoldsmith) December 18, 2017

"(T)he overall investigation has already yielded fruit and there is a clear justification for it to continue," he added.

Just as the FBI's counterintelligence investigation had been looking into contacts between Russian operatives and individuals associated with the Trump campaign prior to Mueller's appointment in May -- the FBI's probe would not necessarily shut down even if Mueller were out of the picture.

"FBI Director (Christopher) Wray would continue to investigate until ordered not to," Goldsmith said.

5/ Pressure in and out of DOJ to appoint Mueller's successor, as opposed to killing investigation, would be enormous, as would pressure to appoint someone of integrity/independence who would continue pushing hard. FBI Dir. Wray would continue to investigate until ordered not to. — Jack Goldsmith (@jacklgoldsmith) December 18, 2017

What's at stake if he does it anyway?

The number two Republican in the Senate has said it would be a "mistake" for Trump to fire Mueller and Democrats would undoubtedly go ballistic if he tried.

Yet the legislative proposals floated over the summer to protect the special counsel have, thus far, stalled.

And even if someone tried to file a lawsuit in federal court to stop the President -- it's difficult to envision a realistic scenario where a challenger with adequate standing would have a legal leg to stand on.

As the head of the executive branch, the President is ultimately in charge of law enforcement.

"The President under our Constitution has the prosecution power. So prosecutors serve as an extension of the Presidency," Katyal has said.

But the very idea that Trump might try to fire or order the firing of the prosecutor -- who now charged the President's former campaign aides with federal crimes and secured a cooperation deal with the President's former national security adviser -- strikes some legal experts as an illustration of the inherent flaw in our constitutional design.

Trump is "a stress test for this particular constitutional problem," Harvard Law School Professor Noah R. Feldman told CNN