Rex Tillerson. AFP

President Donald Trump may look to replace Secretary of State Rex Tillerson with CIA Director Mike Pompeo after an NBC News report said Tillerson had called Trump a "moron" over the summer, according to Mike Allen of the news website Axios.

Trump has addressed the report in several tweets and interviews since the story broke Wednesday morning.

He has gone as far as suggesting that the Senate Intelligence Committee, which is investigating whether he colluded with Russia to meddle in the 2016 election, should investigate "Fake News Networks."

Tillerson gave a special televised address Wednesday in which he disputed the report and reaffirmed his support for Trump and his desire to remain in the administration, but when asked directly about the "moron" comment he dismissed the question as "nonsense" instead of directly denying it.

Now, according to Axios, Tillerson's relationship with Trump is badly damaged. Earlier this week, Ian Bremmer, a political scientist who runs the Eurasia Group, tweeted that Tillerson had taken to deriding Trump's daughter Ivanka and her husband, Jared Kushner, as "the royal family."

Trump has publicly contradicted some of Tillerson's efforts, saying on Twitter that his secretary of state was "wasting his time" trying to talk to North Korea.

But Pompeo, who has defended Trump in the past and speaks with him daily, could seemingly transition smoothly into the role of secretary of state. Pompeo has credibility as a member of Trump's inner circle and, unlike Tillerson, has a wide variety of government experience.

But sources told Axios that Trump's chief of staff, John Kelly, did not want any more staff shake-ups, as they generate negative media coverage.