(CNN) The job of White House chief of staff is close to impossible at the best of times, but President Donald Trump 's new West Wing enforcer faces an even tougher mission -- one that may eventually evolve into an effort to save his presidency itself.

The next chief will walk into a White House engulfed in scandal, in the sights of special counsel Robert Mueller and newly empowered Democrats, at what is shaping up as one of the most grave constitutional moments in US history.

The long-expected departure of the current incumbent, John Kelly, announced by Trump on Saturday , comes at a time of instability and crisis that is remarkable even for this riotous administration where turbulence is the rule.

That left the President without an obvious frontrunner for the toughest job in politics as the White House is being battered by multiple crises, including a trade war with China, turmoil in the markets and a revolt by Senate Republicans over Saudi Arabia policy. And as it braces for a punishing new era of Democratic oversight, Trump's team is also gearing up for the imminent escalation in the President's re-election campaign.

If that was not enough, the new chief of staff will also be faced with the likely hopeless task of trying to tame a President who appears to be deeply rattled by Mueller's strides in recent days, despite his insistence that everything is sunny in the White House.

"The Trump Administration has accomplished more than any other U.S. Administration in its first two (not even) years of existence, & we are having a great time doing it!" Trump tweeted Sunday.

"All of this despite the Fake News Media, which has gone totally out of its mind - truly the Enemy of the People!"

Washington is buzzing with rumors about who Trump could choose to run the West Wing, including the names of budget chief Mick Mulvaney, Treasury Secretary Steven Mnuchin and Rep. Mark Meadows, who chairs the conservative House Freedom caucus

Two sources told CNN that one of the reasons Ayers did not take the job was because of resistance from first lady Melania Trump and some other senior staff.

Such inner White House turmoil is one reason why other candidates might think twice about taking the position.

'Impeachable' offenses

The job of chief staff is often thankless and exhausting even in a conventional administration. And whoever eventually decides to become Trump's third chief will face more than the complications of dealing with a President who constantly torches the conventions of his office and who revels at being the epicenter of an atmosphere of chaos, backstabbing and recrimination.

He or she will also inherit a White House that is more deeply threatened by a criminal investigation and allegations of abuse of power than at any time since the Nixon administration. And every week, the situation gets darker with the President under threat from formidable prosecutors on two fronts -- from Mueller who works under the supervision of the Justice Department and from the US Attorney for the Southern District of New York.

Several weeks of disclosures by Mueller have made it clear that the President and his West Wing are now directly threatened by the investigation.

Therefore, anyone coming into the Trump White House from outside will do so in the knowledge that they are entering a situation that could expose them to reputational, political and even personal legal jeopardy.

The new chief of staff will work for a President who has been accused, effectively by his own Justice Department, of directing and cooperating in the commission of a crime -- in payoffs to women who accused him of affairs, in contravention of campaign finance laws.

On Sunday, Rep. Jerrold Nadler, who will run the Judiciary Committee in the new Democratic House, suggested that the accusations, contained in filings about the case of former Trump lawyer Michael Cohen by the Southern District, imperiled Trump's presidency.

"They would be impeachable offenses. Whether they're important enough to justify an impeachment is a different question," Nadler said on CNN's "State of the Union." "But certainly, they're impeachable offenses, because, even though they were committed before the President became President, they were committed in the service of fraudulently obtaining the office."

While it appears unlikely that Democrats would seek to impeach Trump solely on the basis of campaign finance violations, recent weeks have made clear that Mueller's investigation is broader than was publicly evident and could find worse alleged transgressions by the President and those around him.

For months, the line from the Trump White House has been that there was "no collusion" in a Russian election meddling effort in 2016.

But the special counsel investigation is now uncovering multiple approaches by Russians to the Trump campaign and revealing the openness of key figures to those approaches and their propensity to lie about the contacts.

So there is a genuine possibility that the next chief of staff could find himself presiding over a White House embroiled in an impeachment drama — and leading a war room battling to maintain GOP support for the President in the event that he would go on trial in the Senate.

White House under scrutiny

Just days after the President said there was nothing wrong with pursuing a plan to build a Trump Tower in Moscow while he was a private citizen, Mueller suggested there might be.

From court filings, it appears the special counsel believes that it was relevant that Trump was pursuing a deal with hundreds of millions at stake, which may have needed the help of a foreign power -- Russia -- that was at the same actively interfering in the election and had developed a preference for his candidacy.

And it is now clear that Mueller's probe into alleged obstruction of justice by the President runs far deeper than the events surrounding the firing of former FBI Director James Comey. The revelation that Mueller recently interviewed Kelly, reported by CNN, as well as last week's court documents, make clear that activity in the White House in 2017 and 2018 is also under scrutiny. That's because Kelly did not even take the job until July 2017 -- two months after Comey was fired.

Mueller has so far not made any direct accusations against the President or senior officials. But embedded in recent court filings is the possibility that he is looking at whether there were efforts by the administration to thwart investigations by his team and Congress.

For instance, in a sentencing memo, Cohen's lawyers said their client was in "close and regular contact" with White House staff and Trump's lawyers ahead of congressional testimony in which Cohen lied by saying the proposed Moscow project was off the table by the time of the Iowa caucuses in 2016.

Mueller's filings also note that Trump's former campaign chairman, Paul Manafort, was in touch with senior officials even after he was indicted.

This all makes it clear that after months when Mueller was dealing with Trump associates from his past life as a businessman and on the campaign, his investigation is now penetrating the White House itself and is interested in events that took place long after the campaign ended.

Trump's defenders correctly argue that Mueller has yet to directly accuse the President of any criminal actions. And it is often overlooked that Mueller's filings are the work of a prosecutor who by definition is making the best possible presentation of a case not yet challenged in court.

But the direction of Mueller's investigation appears clear. He is getting closer and closer to the President and his inner circle.

And sooner or later it will be up to the next chief of staff to navigate the White House through a crisis that could put his boss's presidency in existential danger.