Washington (CNN) If Donald Trump really thinks the planet is a "mess" as crises churn from North Korea to the Middle East, it would seem illogical to ditch a secretary of state he once praised as a "world-class player."

Yet Rex Tillerson is on the thinnest of ice, estranged from a President who is now furious with his top diplomat, amid reports that the courtly Texan broke character and branded him "a moron" after a meeting this summer

Tillerson's failure to deny the NBC report in a press conference on Wednesday may have finally severed his relationship with Trump, but his position has been weakening for months, despite his denials that he has been close to resigning.

The President and senior staff members appreciate the poor political optics of yet another important official being shown the door, sources have told CNN. And ditching the secretary of state now could complicate Trump's reorientation of Iran policy expected to take place next week with his decertification of a nuclear deal. Tillerson has also been preparing Trump's tour of Asia in November.

So Tillerson may cling on a little longer on the political equivalent of death row. But when his apparently inevitable departure does occur, it will come at a cost, both to America's already battered credibility overseas and to the cause of stable government in Washington.

Abroad, Tillerson's departure could deepen confusion about US foreign policy and doubts about the administration's competence. At home, it may dismay foreign policy graybeards like Sen. Bob Corker who said this week Tillerson was one of a group including Defense Secretary James Mattis and Chief of Staff John Kelly, keeping the US "from chaos." Establishment figures also worry that a vacancy at State could provide an opening for a hawk, someone like former UN envoy John Bolton, who might indulge Trump's wilder impulses.

There is a critique common to many senior foreign officials and diplomats about the administration -- they don't know who is running the show, understand its foreign policy strategies and are uncertain who to call in a crisis.

Losing a secretary of state so quickly would exacerbate those concerns.

"It's difficult to find out who is in charge or who to call on the phone," said a senior official from one key US ally recently.

North Korea, Iran crises

Upheaval in the national security team would also raise doubts about how Washington will handle several worsening crises.

Tillerson has sought to keep open a channel of communication with North Korea amid a nuclear showdown exacerbated by Pyongyang's missile launches and Trump's incendiary rhetoric. The President, however, undermined his secretary of state last week by ordering him over Twitter to stop wasting his time in talking to Pyongyang.

I told Rex Tillerson, our wonderful Secretary of State, that he is wasting his time trying to negotiate with Little Rocket Man... — Donald J. Trump (@realDonaldTrump) October 1, 2017

Tillerson has also been the most important US conduit with Russia, trying to stop a vital relationship between two nuclear powers from getting even worse.

A senior administration official last month paid tribute to Tillerson's expertise on Russia, developed during his many business trips to the country as the CEO of ExxonMobil.

With tensions at their highest since the end of the Cold War, with US and Russian troops in proximity on Syria's battlefields, Tillerson manages a channel of communication with Moscow that can be instantly activated in a crisis, the official said, speaking on condition of anonymity.

Tillerson was the only US official to accompany Trump to his meeting with Russian President Vladimir Putin in Germany in July. And given the political constraints imposed on the White House by the special counsel probe into allegations of election collusion between the Trump campaign and the Kremlin, solid State Department links with Russia are even more important than usual.

Yet despite the disruption his departure would cause, Tillerson hasn't exactly been blazing a trail of glory. US allies have been outraged by Trump administration policies, including the withdrawal from the Paris accord. It's quite possible he would hardly be missed. And his position appears untenable.

"Rex Tillerson has been dealt a bad hand by the Potus & has played it badly. For both reasons he cannot be effective SecState & should resign," Richard Haas, President of the Council on Foreign Relations wrote on Twitter on Thursday.

There's no doubt Trump has made Tillerson's job harder. For instance, in June, Tillerson called on Qatar's neighbors to ease a blockade against the US ally. Hours later, the President backed the position of Saudi Arabia and Bahrain that Qatar was guilty of sponsoring terrorism.

Trump has also eroded Tillerson's authority by farming out State Department functions, for instance putting his son-in-law Jared Kushner, who has no diplomatic experience, in charge of a Middle East peace initiative.

Philip Zelikow, a State Department counselor under former secretary of state Condoleezza Rice, said that "observant foreign governments have already formed their impressions about the President and about the secretary."

The consequences of their latest spat are "likely to be more internal, within the administration, adding that much more acid to the corrosive drip," he said.

Tillerson has underwhelmed in Washington just as he has abroad. He's not seen as a fighter for his department on Capitol Hill, a factor that has deepened a crisis of morale at Foggy Bottom. He has seemed more fixated on his plan to cut the department's budget than global strategy. His hermit-like approach to the press has undermined US public diplomacy. The result is that Tillerson lacks the domestic power base and strong persona that could insulate him from Trump's capricious nature.

"Tillerson has been in a difficult situation that he has made worse by the way he has conducted himself in office," said Derek Chollet, a former senior defense and foreign policy official in the Obama administration now with the German Marshall Fund. "Contrast him with someone like Mattis, who I think has a similar disposition in terms of his views on the US role in the world."

Whoever is to blame, Tillerson's disconnect with Trump is especially damaging.

A Secretary of State's authority rests on his relationship with the President. As it stands, the rest of the world believes that Tillerson thinks his boss is a moron and that Trump doesn't trust his top diplomat. That being the case, the crucial bond between the two already looks irrevocably fractured.

Yet any new secretary of state would confront the same, possibly insurmountable challenge that Tillerson has experienced -- the President himself.

Tillerson's resume, character and approach would probably equip him just fine to work in a more conventional administration.

The question his difficult tenure raises is whether Trump would allow any secretary of state who serves him the room to act, or whether even more fundamentally, the President actually believes in diplomacy itself.