“I have always followed the rules and I hope such a review will provide further clarity,” Mr Sunak said on Twitter.
In his letter to the prime minister, Mr Sunak said his “overriding concern” was to retain public confidence in the answers that were given.
The move followed accusations by Labour of a conflict of interests after the disclosures that his wife was “non-domiciled” for tax purposes and that he retained a US green card while Chancellor.
Also on Sunday, a government minister said Mr Sunak has been a “remarkable force for good” in British politics, as Labour continued to press for him to come clean about his family’s tax affairs.
Policing Minister Kit Malthouse acknowledged the disclosure that Mr Sunak’s wife, Akshata Murty, was “non-domiciled” in the UK for tax purposes was “not ideal”.
But Mr Malthouse said the chancellor was a “smart, clever, committed politician” who had helped to steer the economy through the worst of the pandemic.
Mr Sunak had ordered a Whitehall inquiry into who leaked details of Ms Murty’s tax status to the media, causing a furious outcry.
Labour's shadow home secretary Yvette Cooper said it was a matter of “basic fairness” that Mr Sunak’s family should pay tax in the same way as everyone else, and should not enjoy “special arrangements” to reduce their bills.
“I don’t think the Chancellor gets that," Ms Cooper told BBC1. “That raises real questions about his judgment."
Ms Murty, who remains an Indian citizen, announced on Friday that she would now pay UK taxes on all of her worldwide income, because she did not want her tax status to be a distraction for her husband.
It has been estimated that her non-domiciled status could have saved her £20 million ($26m) in taxes on dividends from her shares in Infosys, an Indian IT company founded by her father.
Mr Malthouse said it was “not a brilliant time” for the details to come out when the country was struggling with a cost-of-living crisis, but said she had now “corrected” the situation.
With some Conservative MPs questioning whether any hopes Mr Sunak harboured of becoming prime minister have been dashed by the disclosures, Mr Malthouse denied his career was “toast”.
“Rishi Sunak has been a remarkable force for good in this country over the last two years," he said. "He put in place some incredible support schemes during the pandemic at enormous speed."
But Mr Malthouse said that ultimately, Mr Sunak’s political future was of “secondary importance” to what happened to the UK economy.
“It is not ideal, but we all when we sign up for this job know that there is going to be rough times and smooth times,” he told the BBC.
“The key thing is that he remains committed to the job that he is doing, which is to steer the British economy and its people through some very, very difficult and challenging moments.
“What happens to him the future will be of secondary importance to what happens to the country.”
There is likely to be relief among Mr Sunak’s allies at the way Mr Malthouse, a longstanding ally of Mr Johnson, spoke out in his support.
The chancellor was reportedly unhappy at the failure of colleagues to come to his defence when the news broke last week.
Mr Sunak was already under pressure amid criticism that last month’s spring statement did little for those struggling on low incomes, leading to reports of tension with Downing Street.
Meanwhile, sources close to Mr Sunak have played down reports that he was moving his wife and two daughters out of Downing Street to escape the glare of the media.
The Sunday Times reported that removal vans arrived in the street on Saturday to take the family's belongings away.
But one source said they had always intended to spend more time in their west London home as their elder daughter, Krishna, entered her final term of primary school.
Mr Sunak is still expected to stay in Downing Street during the week while joining them at weekends.
Meanwhile, Health Secretary Sajid Javid has disclosed that for six years while working as an international banker and before entering politics he was classified as non-domiciled.
“For some of those years I was non-domiciled for tax purposes, but I paid all UK taxes due on my income and have always done so,” Mr Javid told The Sunday Times.
“In 2009, upon my return to the UK, I became tax resident in the UK again and also proactively chose to give up my non-domiciled status by making the UK my ‘domicile of choice’.”
He said he also benefited from an offshore trust for several years but on becoming a minister he decided voluntarily to close it, repatriate all assets to the UK and pay 50 per cent tax on the dividends.
“This approach deliberately incurred the heaviest possible tax burden and offset any accrued benefits from the previous trust arrangement, but I believed it was the right thing to do,” Mr Javid said.