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Rishi Sunak came under pressure at Prime Minister's Questions over former Chancellor Nadhim Zahawi's tax affairs.
Mr Zahawi is reported to have paid more than £1m to Revenue and Customs to settle a tax dispute.
He has so far declined to say whether the story, in The Sun, is correct.
Mr Sunak said the minister, who is now Conservative Party chairman, had "already addressed this matter in full", in response to a Labour MP's question.
Mr Zahawi is facing questions from opposition parties about whether he tried to avoid paying tax by using an offshore company to hold shares in YouGov, the polling company he co-founded.
A spokesman for Mr Zahawi has said his taxes are "properly declared" and that he "has never had to instruct any lawyers to deal with HMRC on his behalf".
"He is proud to have built a British business that has become successful around the world," the spokesman added.
At PMQs, Labour MP Alex Sobel said Mr Zahawi was "forced to pay millions to HMRC to settle a tax dispute".
He asked Mr Sunak if he was "aware of an investigation when he appointed him to his cabinet and as chairman of the Conservative Party".
He added: "Will the prime minister demand accountability from his cabinet members about their tax affairs?"
The prime minister said Mr Zahawi "has already addressed this matter in full and there's nothing more that I can add".
Mr Sunak's official spokeswoman said Mr Zahawi "has spoken and been transparent with HMRC".
On whether Mr Sunak believes the matter is now closed, she said: "I don't know whether the prime minister has reviewed it in full, but I do know that he takes Nadhim Zahawi at his word."
Asked if Mr Sunak is confident he knows everything he needs to know, she responded "yes" and said the prime minister had full confidence in Mr Zahawi.
Home Office minister Robert Jenrick said it was a "private matter" for Mr Zahawi adding the "important factor" is that Mr Zahawi's tax affairs are now up to date.
The BBC has repeatedly asked Mr Zahawi's representatives if he has paid a tax bill to settle a dispute with HM Revenue and Customs but has yet to receive an answer.