The White House is discussing plans to replace Secretary of State Rex Tillerson with CIA chief Mike Pompeo, US media say.
Mr Tillerson has been at odds with Mr Trump over foreign policy recently.
The secretary of state was even reported to have privately described the president as a “moron”.
They have differed in their approach to North Korea’s missile testing and Iran’s nuclear programme, among other issues.
But the White House has said Mr Tillerson continues in office; and the state department said “the rumours” were not true.
The reports emerged initially in the New York Times and Vanity Fair, quoting government sources.
Associated Press quotes two unnamed White House officials as saying a plan is being discussed.
Mr Pompeo would be replaced at the CIA by Republican Senator Tom Cotton, the New York Times reports.
Reports suggest the change could take place as soon as December or in January.
However, it is not yet clear whether Mr Trump has given final approval to the move, the NYT says.
What has the official reaction been?
The White House has said Mr Tillerson remains in post.
“As the president just said: ‘Rex is here,'” White House spokeswoman Sarah Sanders said in a statement.
“There are no personnel announcements at this time.
“Secretary Tillerson continues to lead the state department and the entire cabinet is focused on completing this incredibly successful first year of President Trump’s administration.”
Briefing reporters later, state department spokeswoman Heather Nauert admitted that Mr Tillerson and Mr Trump have had policy differences.
But she said that Chief of Staff John Kelly had phoned the department to say “the rumours are not true.”
When asked how Mr Tillerson, who is due in Europe next week, could continue to do his job while the White House was briefing he was about to lose it, she said: “The secretary of state is someone who doesn’t let his feathers get ruffled very easily.”
In further reaction to the reports, Senate Foreign Relations Committee Chairman Bob Corker said Mr Tillerson was “unaware of anything changing”.
Why would Trump want Tillerson gone?
They don’t appear to get on.
Mr Trump’s disenchantment with Mr Tillerson, a former chief executive of energy giant Exxon Mobil, has been rumoured for some time.
The secretary of state has defended the multi-party deal to curb Iran’s nuclear ambitions in return for a loosening of sanctions – an agreement derided by Mr Trump.
They have also differed, at least publicly, on North Korea.
The president said in October that Mr Tillerson was “wasting his time” trying to bring about diplomatic contacts with Pyongyang as it continues to build up its missile and nuclear capabilities.
The “moron” comment is unlikely to have helped relations either.
And in June, Mr Trump and Mr Tillerson were giving out contradictory messages about the dispute between Saudia Arabia and Qatar.
The secretary of state warned that the Saudi-led blockade of Qatar could affect the fight against extremism and was having humanitarian consequences.
Mr Trump, on the other hand, appeared to endorse Saudi policy, suggesting the blockade might herald “the beginning of the end to the horror of terrorism” – a reference to allegations Qatar allows funding of extremist groups.
To add to Mr Tillerson’s woes, his plans for radical restructuring of the state department are not going down well in some quarters.
His announcement that he would “create the state department of the future” involves sacking career diplomats, and budget cuts.
Democrats, and some Republicans, have expressed concern the restructuring could undermine America’s interests abroad.
Analysis: Failure to navigate Washington politics
By Barbara Plett, BBC State Department correspondent
Tensions between Mr Trump and Mr Tillerson have been widely reported. That seems partly for personal reasons – the two are chalk and cheese in temperament and the way they work. But Mr Trump is also said to have complained that his secretary of state is too “establishment”.
The president has undermined Mr Tillerson on key policy areas with his fiery tweets. Yet he’s also quietly accepted the secretary’s diplomatic strategy on issues like North Korea, suggesting the relationship is more complex than reported.
Separately, Mr Tillerson’s management style has also alienated chunks of his natural constituency: many in the state department, Congress, and foreign policy community. That’s because he accepted Trump’s proposed budget cuts without much of a fight and, at the same time, launched a major redesign of the department without saying where it was heading or explaining the details.
There’s also been an exodus of dismissed or disillusioned senior staff who aren’t being replaced quickly enough. Not all of this is Mr Tillerson’s fault, but he has failed to navigate the push and pull of Washington’s politics.
What are the possible wider changes?
The secretary of state’s firing would form part of a wider national security team shake-up overseen by Chief of Staff Kelly, the NYT reports.
Mr Pompeo, as Mr Tillerson’s possible replacement, is reputed to be closer to Mr Trump on security issues.
And Tom Cotton, the Arkansas senator and reported choice for new CIA head, holds similar views to the president on the Iran nuclear deal.
The decorated Iraq and Afghanistan war veteran is described as a hawk who wants to increase the defence budget.
Mr Tillerson, 65, was appointed in January and his sacking would cap one of the shortest tenures for an American secretary of state.
His appointment was a controversial one, thanks to his close links to Russian President Vladimir Putin stemming from his work on major oil contracts in Russia.
The Trump administration faces several investigations into whether there was collusion between his election campaign and Russian officials.
Mr Tillerson was on the verge of retirement when he was asked to fill the role.
Interviewed earlier this year by the conservative website, Independent Journal Review (IJR), he was quoted as saying: “I didn’t want this job. I didn’t seek this job… my wife told me I’m supposed to do this.”