The state department has said it has been assured by the White House there are no plans to oust Rex Tillerson and replace him with the CIA director, Mike Pompeo.

Donald Trump himself was less reassuring, however. When asked whether Tillerson would stay in his job, the president simply noted that the secretary of state was at the White House on Thursday.

“He’s here. Rex is here,” Trump said.

The cryptic non-denial added fuel to multiple reports quoting White House officials as saying a plan was under consideration to get rid of Tillerson, put Pompeo, a former Republican congressman in his place, and make Senator Tom Cotton, another Trump loyalist, CIA director.

The reports of a reshuffle among top foreign policy and national security officials came against a backdrop of stories of clashes between Trump and Tillerson. According to the New York Times, the first to report an impending departure, the plan was being put together by the White House chief of staff, John Kelly.

But state department spokeswoman Heather Nauert said that Kelly had called after the news reports broke and insisted they were false.

“What I can tell you is that chief of staff Kelly called our department this morning and said that the rumours are not true, that those reports are not true. That is what I’ve been told,’ Nauert said.

As for Tillerson, she said: “He remains, as I have been told, committed to doing this job. He does serve at the pleasure of the president. This is a job that he enjoys. He is continuing with his meetings. He’s continuing with his calls.”

She pointed out that Tillerson had a full week of engagements next week when he tours Europe, going to a Nato foreign ministers’ meeting in Brussels, then to Vienna and Paris. Asked whether the swirl of rumours undermined his authority as an administration representative abroad, Nauert said: “The secretary is someone whose feathers don’t get ruffled very easily.”

She acknowledged Tillerson and Trump had “areas of disagreement when it comes to policy”, but that the president respected Tillerson and valued his opinions. Nauert insisted they had a “cordial relationship”, but added: “Where that relationship is today, I can’t speak to that.”

Tillerson met defence secretary James Mattis, his closest ally in the administration, early on Thursday morning, the state department said. Later in the day, Mattis was asked what he makes of reports of Tillerson leaving.

Mattis replies: “I make nothing of it, there’s nothing to it.”

Sarah Sanders, the White House press secretary, stopped short of saying Tillerson has a long-term future in the administration. “When the president loses confidence in someone, they will no longer serve in the capacity that they’re in,” she told reporters. “The president was here today with the secretary of state. They engaged in a foreign leader visit and are continuing to work together to close out what we’ve seen to be an incredible year.”

Sanders denied that the media reports are making it difficult for Tillerson to do his job. “The secretary of state’s a pretty tough guy,” she said. “I think he’ll be just fine carrying his job out.”

The president’s relations with Tillerson, a former oil executive, have been increasingly strained. The secretary of state publicly disowned him over Trump’s response to a neo-Nazi march in Virginia, saying the president “speaks for himself”.

Tillerson was also reported to have referred to Trump as a “moron” at a meeting of national security officials, a report he did not deny.

For his part, Trump has repeatedly contradicted and undermined Tillerson’s attempts at diplomacy, telling him on Twitter in October that he was “wasting his time” seeking a dialogue with the North Korean regime.

The Republican chair of the Senate foreign relations committee, Bob Corker, told reporters on Thursday morning that he had just talked to Tillerson, who had not been informed of any forthcoming changes.

“He’s conducting business, as is the norm, and is unaware of anything changing,” Corker said. Corker, the Tennessee senator, who recently said that Tillerson and a few other top officials “help separate us from chaos”, suggested that the secretary of state was the target of palace intrigue.

“It’s been evident to me that for some time, somebody is seeking to undermine his presence there. I don’t know who that is,” he said.

A spokeswoman for Senator Cotton insisted his “focus is on serving Arkansans in the Senate”.

If the changes do take place as reported, the immediate effect would be a cabinet that was on balance more personally loyal to Trump, and more hawkish, particularly on Iran. Pompeo and Cotton have been staunch opponents of the 2015 nuclear deal with Tehran. Cotton has been a longstanding advocate of military action, suggesting limited airstrikes would be enough to contain the Iranian nuclear programme.

“The United States has the ability to totally destroy Iran’s nuclear infrastructure,” Cotton said in October. “If they choose to rebuild it we can destroy it again.”

Pompeo has also suggested the administration was considering pursuing regime change in North Korea, something that Tillerson has publicly ruled out.

Thomas Wright, the director of the centre on the United States and Europe at the Brookings Institution, said he doubted Trump, a former reality show host, would announce all the personnel changes in one go.

“I would be surprised if he did it immediately. He loves the theatre of doing auditions and then having people wait for the announcement of the winner of the show,” Wright said.

Ivo Daalder, a former Nato ambassador and president of the Chicago Council on Global Affairs, said: “A decision by President Trump to remove Rex Tillerson as secretary of state would be unprecedented. It’s been more than a century since a secretary of state has been fired.”