United States (US) Secretary of State Rex Tillerson firmly denied Wednesday that he had fallen out with Donald Trump and vowed to remain in post, dismissing a report that he had once dubbed the president a "moron."

For his part, Trump declared he has "total confidence" in his top diplomat, after both had denied an explosive report that Tillerson became so frustrated over the summer that he considered resigning.

According to the NBC News story, which cited "multiple senior administration officials" but was described as "erroneous" by the State Department, Tillerson had referred to Trump as a "moron" at a July 20 Pentagon meeting.

Afterwards, the report said, Tillerson met with Vice President Mike Pence, who urged him to show more respect, and with other senior officials who urged him not to resign.

Appearing before reporters at a hastily organised news conference in the State Department, Tillerson denied the report and pledged Trump his full support.

"There's never been a consideration in my mind to leave," the former oil executive said.

"I serve at the appointment of the president and I am here for as long as the president feels I can be useful to achieving his objectives."

Tillerson's spokeswoman Heather Nauert later said the secretary had called Trump.

"He told me that, Heather, it was quote 'a good conversation' and that 'we're all good,'" she said.

The former ExxonMobil chief executive's tenure at the State Department has drawn scorn from Trump's opponents, from former diplomats and from the Washington policy elite.

He has also been faced with an extraordinary array of foreign policy challenges ─ from North Korean nuclear threats to Russian subversion to attacks on US diplomats in Cuba.

Taunting tweets

But his efforts have been overshadowed ─ some would say undermined ─ by Trump's un-diplomatic style and his streams of taunting tweets stirring international tensions.

Formerly a respected figure in the oil business, the 65-year-old Texan has been attacked for failing to develop his own political profile and for failing to fight the State Department's corner.

It is commonplace for him to be described by Washington pundits as the weakest secretary ever ─ and one scathing Washington Post column this week called him "Donald Trump's dog."

But his public loyalty to Trump hasn't always been rewarded with support from the top.

On Sunday, as Tillerson flew home from meeting with top Chinese officials, Trump tweeted that his envoy was "wasting his time" in trying to probe North Korea's willingness to talks.

The State Department denied this was a reprimand, insisting Trump had been warning Kim Jong-Un that he should respond quickly to diplomatic overtures or face tougher action.

And, on Tuesday, Tillerson's cabinet ally Defense Secretary Jim Mattis pointedly told lawmakers at a congressional hearing that he backed the secretary's North Korea strategy.

But Trump's Twitter outburst had triggered a new round of speculation.

Richard Haass, former chief policy adviser to then secretary Colin Powell and the 15-year president of the influential Council on Foreign Relations, said Tillerson should go.

"Rex Tillerson has been dealt a bad hand by the Potus and has played it badly. For both reasons he cannot be effective secretary of state and should resign," he tweeted Wednesday.

Tillerson came out fighting.

He did not himself directly address the damaging allegation that he had called Trump a "moron" ─ but Nauert later stated that the secretary "did not use that type of language... he did not say that."

Tillerson alleged that unidentified others were spreading malicious rumors to tear down the president's agenda, and vowed: "I do not and I will not operate that way."

'Categorically false'

He praised his cabinet colleagues, cited their main achievements, and promised: "There's much to be done and we're just getting started."

And he spoke warmly of Trump.

"He loves his country. He puts Americans and America first. He's smart. He demands results wherever he goes," Tillerson said.

Trump tweeted his support immediately after Tillerson's remarks and later told reporters: "It was fake news. It was a totally phoney story."

The vice president's office also issued a statement insisting Pence had never discussed Tillerson's possible resignation, stating: "Any reporting to the contrary is categorically false."