The SNP's Westminster deputy leader, Kirsten Oswald, said that Mr Sunak’s position as chancellor was “increasingly untenable” as questions still remained over his tax payments.
His spokesman has previously said that he has paid taxes “where required” over the period.
It comes as a full Whitehall inquiry got underway into who leaked details about his wife's tax arrangements. Akshata Murty announced on Friday that she would pay UK taxes on her worldwide income after the disclosure she was “non-domiciled” in the UK for tax purpose sparked a political storm. Non-dom status allows someone to pay no UK tax on their overseas earnings.
It has been estimated that her non-dom status could have saved her £20 million in taxes on dividends from her shares in Infosys, an Indian IT company founded by her father.
Mr Sunak’s team is said to believe that it is a Labour-supporting official dubbed “red throat” who is behind the leak to the media.
Meanwhile, reports claimed that Sajid Javid claimed non-domicile tax status for six years when he was a banker. Mr Javid admitted that he had been a ‘non-dom’ between 2000 and 2006, when he was working for Deutsche Bank, having qualified for the scheme because his father was born in Pakistan.
Mr Javid said: “I have been domiciled in the UK for tax purposes throughout my entire public life. Given heightened public interest in these issues, I want to be open about my past tax statuses.”
He added: “My career before politics was in international finance. For almost two decades I constantly travelled around the world for work.”
He said that after living in New York for a time, he returned to the UK. He said: “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.”
It emerged last week that Mr Sunak held a US green card until October 2021 - despite becoming Chancellor a year earlier in February 2020. Tax experts have suggested that the arrangement may actually have led to the chancellor paying more tax, filing returns in both the UK and US, but calls for clarification are growing.
Ms Oswald said: "Rishi Sunak's response to revelations that he held a US green card for a significant period of time when he was an MP and Chancellor raises more questions than answers. It is now vital for transparency and trust that the Chancellor publishes his tax returns over the period he held a US Green Card since he has been an MP to set out whether or not he avoided paying more tax in the UK.
"It is staggering that Mr Sunak is hammering ordinary families across the UK with tax hikes, refusing to tackle the Tory cost-of-living crisis, and gifting bankers with a £4 billion tax cut, all the while his billionaire family are saving millions through a tax loophole and he has been found to have held a US green card - with all the tax implications that come with that. Crucially, we need to know whether Mr Sunak failed to pay UK tax while serving as Chancellor of the Exchequer."
She added: “His spokesperson has said tax was paid ‘where required’ for the duration he held his Green Card. Does that mean he paid US but no UK tax for a period while living in Downing Street and setting tax for all other UK citizens? If so, it is an astonishing admission and makes his position as Chancellor increasingly untenable."
Speaking on TV on Sunday morning, policing minister Kit Malthouse insisted that Mr Sunak has been a "remarkable force for good" in British politics.
Mr Malthouse acknowledged the disclosure that Mr Sunak's wife, Akshata Murty, was "non domiciled" in the UK for tax purposes was "not ideal".
However he said the Chancellor was a "smart, clever, committed politician" who had helped steer the economy through the worst of the pandemic.
"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," he said.
Ultimately, however, he said Mr Sunak's own 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 said. "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."
For Labour, shadow home secretary Yvette Cooper said it was a matter of "basic fairness" that the Chancellor'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,” she said. "That raises real questions about his judgement.”
The Chancellor, who has previously been tipped as a likely successor to Prime Minister Boris Johnson, was reportedly unhappy at the failure of colleagues to come to his defence when the news broke last week.
He was already under pressure amid criticism that last month's spring statement did little for those struggling on low incomes, leading to reports of tensions within No 10.
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.