Boris Johnson has pulled out of a scheduled appearance before a panel of senior MPs, saying he has to "focus on delivering Brexit".
The prime minister had been due to be grilled by the Commons liaison committee - made up of the chairs of select committees - on Thursday.
He has asked committee chair Sarah Wollaston for a date "five or six months" from him becoming PM.
Dr Wollaston accused the PM of "refusing to face detailed scrutiny".
The former Conservative - now Lib Dem - MP said it was the third time the prime minister had cancelled.
In a handwritten note, the PM said: "Dear Sarah, I promised that I will come to the Liaison Committee and I will keep that promise but I am afraid I must now focus on delivering Brexit in the difficult circumstances in which we now find ourselves, and I believe I would be of greater value if I could postpone to a fixed date nearer 5 or 6 months after I became PM, so that my appearance took place after roughly the same period in office as Gordon Brown, David Cameron & Theresa May.
"I do hope you will understand."
Dr Wollaston, who quit the Conservatives over Mr Johnson's Brexit policy, expressed "astonishment" that he had cancelled at such short notice.
In her reply to the PM she said: "You are refusing to face detailed scrutiny from select committee chairs tomorrow morning.
"This is the third time that you have postponed or cancelled."
She added: "Our role as select committee chairs is to ask you detailed questions on behalf of the public and we planned to do so on Brexit, climate change, health and social care. It is unacceptable that you are refusing to be held to account."
Dr Wollaston released a copy of the PM's handwritten note to the media.
She also raised the issue in the Commons, where she was joined by other committee chairs - Labour MPs Yvette Cooper and Mary Creagh - in accusing him of avoiding accountability.
But Conservative MP Sir Patrick McLoughlin, a member of the liaison committee, defended the prime minister, saying he "is held to account in this chamber by all members of Parliament" at prime minister's questions.
Another Conservative MP, Steve Baker, took a swipe at Dr Wollaston, saying it was a "profound injustice" that some MPs were able to keep their "high office" in the select committee system despite abandoning their party.
Dr Wollaston hit back, saying committee chairs were elected by the "whole House of Commons" because they were "trusted" to be impartial.
Deputy Speaker Lindsay Hoyle said he hoped the PM would appear before the committee "at the earliest possible date", which he said meant "this year, not next year" - unless "other events overtake us".