"I asked John for his resignation, which was given to me this morning," Mr Trump tweeted, saying he will appoint a replacement next week.
But Mr Bolton insisted he had quit, and vowed to have his say "in due course".
Appointed in April 2018, he was Mr Trump's third national security adviser after Michael Flynn and HR McMaster.
White House press secretary Stephanie Grisham told reporters: "[The president] didn't like a lot of his policies, they disagree."
She said the president had asked for Mr Bolton's resignation on Monday night, and it was delivered on Tuesday morning.
What is Bolton saying?
But moments after Mr Trump's tweet, Mr Bolton took to Twitter to offer a different version of events.
He maintained he had actually offered his resignation, but Mr Trump told him "let's talk about it tomorrow".
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