US President Donald Trump was "well within his rights" to fire National Security Adviser John Bolton, according to US Secretary of State Mike Pompeo.

The diplomat and US Secretary of Treasury Steven Mnuchin faced repeated questions about the fallout from Bolton's acrimonious exit from the White House this Tuesday.

"The president is entitled to the staff that he wants, at any moment," said Pompeo.

"He should have people that he trusts and values and whose efforts and judgements benefit him in delivering American foreign policy."

Trump announced the move on Twitter, saying the pair "disagreed strongly".

However, Bolton insisted he quit and vowed to have his say "in due course".

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