Advisers and allies of President Trump have suggested replacing Secretary of State Rex Tillerson Rex Wayne TillersonGary Cohn: 'I haven't made up my mind' on vote for president in November Kushner says 'Alice in Wonderland' describes Trump presidency: Woodward book Conspicuous by their absence from the Republican Convention MORE with CIA Director Mike Pompeo Michael (Mike) Richard PompeoHouse panel halts contempt proceedings against Pompeo after documents turned over Outgoing ambassador to China slams Beijing over coronavirus: 'Could have been contained in Wuhan' Hillicon Valley: FBI chief says Russia is trying to interfere in election to undermine Biden | Treasury Dept. sanctions Iranian government-backed hackers MORE, according to an Axios report.

Trump is said to have a good rapport with Pompeo, who delivers the President's Daily Brief. According to Axios, Pompeo would be seen by world leaders as a member of Trump's inner circle in a way Tillerson is not.

At the same time, however, White House chief of staff John Kelly John Francis KellyMORE has reportedly urged the president against high-level shake-ups before next year, fearing that it would only add to the chaos in the administration.

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Trump and Tillerson's relationship has appeared increasingly fraught. The president has publicly undercut his top diplomat more than once, and a bombshell Washington Post report this week portrayed tensions based not only on major policy disagreements, but small matters as well.

In an unusual news conference on Wednesday, Tillerson sought to dispel an NBC News report that White House officials, including Vice President Pence, had to talk him out of stepping down over the summer.

"The vice president has never had to persuade me to remain as secretary of State because I have never considered leaving," Tillerson said.

But in that same news conference, Tillerson did not address part of that report claiming that he had once called Trump a "moron" after a tense meeting over the administration's Afghanistan strategy.

While State Department spokeswoman Heather Nauert later said that Tillerson had assured her he never used such language to describe the president, Axios reported Friday that Trump was angered that the secretary of State did not address the matter himself.