In the UK, the Public Accounts Committee (PAC) plans to investigate the scandal and will require HSBC's former head to give evidence.
There have also been calls for action in the US, Belgium, France, Argentina and Switzerland itself.
HSBC said it is "co-operating with relevant authorities".
The BBC learnt that HSBC helped wealthy clients evade hundreds of millions of pounds worth of tax.
Panorama has seen accounts from 106,000 clients in 203 countries, leaked by whistleblower Herve Falciani in 2007.
In the US a member of the Senate banking committee has asked the government to reveal what it knew about the documents.
In Belgium, a judge is considering issuing international arrest warrants for directors of the Swiss division of the bank, while France has launched an investigation and its prime minister has promised more action both at home and at a European level.
Swiss politicians are asking for regulatory investigation.
The PAC's chairman, Margaret Hodge, said late on Monday: "The Public Accounts Committee will be launching an urgent inquiry to which we will require HSBC to give evidence - and we will order them if necessary".
The PAC has a routine meeting scheduled on Wednesday at which the head of HMRC, Lin Homer, will appear.
Peer pressure
One of HSBC's most senior figures, Stephen Green, who was made group chief executive in 2003 before going on to become the group executive chairman of the bank in 2006, was made a minister eight months after HMRC had been given the leaked documents from his bank. He served as a minister of trade and investment until 2013.
Now Lord Green, he told the BBC's Panorama programme: "As a matter of principle I will not comment on the business of HSBC past or present."
But Mrs Hodge said: "Either he didn't know and he was asleep at the wheel, or he did know and he was therefore involved in dodgy tax practices.
"Either way he was the man in charge and I think he has got really important questions to answer."
John Mann, a Labour MP who sits on the Treasury Select Committee, has written to the chairman of the committee, Andrew Tyrie, and asked him to recall Lord Green: "I've asked for him to be brought in front of the committee so he can put the record straight. If we call him he'll have to come."
Treasury minister David Gauke defended Lord Green's appointment on BBC's Radio 4. "I am not aware of any evidence that suggests that Lord Green was involved in this sort of activity", but said he did not know whether anyone asked him about HSBC prior to his government appointment.
'Control failures'
HSBC admitted that it was "accountable for past control failures." But it said it has now "fundamentally changed".
"We acknowledge that the compliance culture and standards of due diligence in HSBC's Swiss private bank, as well as the industry in general, were significantly lower than they are today," it added.
The bank now faces criminal investigations in the US, France, Belgium and Argentina, but not in the UK, where HSBC is based.
HSBC said it is "co-operating with relevant authorities".
Offshore accounts are not illegal, but many people use them to hide cash from the tax authorities. And while tax avoidance is perfectly legal, deliberately hiding money to evade tax is not.
The French authorities concluded in 2013 that 99.8% of their citizens on the list were probably evading tax.
Joint investigation
The thousands of pages of data were obtained by the French newspaper Le Monde. In a joint investigation, the documents have now been passed to the International Consortium of Investigative Journalists, the Guardian newspaper, Panorama and more than 50 media outlets around the world.
HM Revenue and Customs (HMRC) was given the leaked data in 2010 and has identified 1,100 people from the list of 7,000 British clients who had not paid their taxes. One person has been prosecuted.
HMRC said £135m in tax, interest and penalties have now been paid by those who hid their assets in Switzerland.
But Ms Hodge said: "I just don't think the tax authorities have been strong enough, assertive enough, brave enough, tough enough in securing for the British taxpayer the monies that are due."
In some cases, HSBC broke the law by actively helping its clients.
The bank gave one wealthy family a foreign credit card so they could withdraw their undeclared cash at cash dispensers overseas.
HSBC also helped clients stay ahead of the law.
When the European Savings Directive was introduced in 2005, the idea was that Swiss banks would take any tax owed from undeclared accounts and pass it to the taxman.
It was a tax designed to catch tax evaders. But instead of simply collecting the money, HSBC wrote to customers and offered them ways to get round the new tax.
HSBC denies that all these account holders were evading tax.
Verbal messages
Meanwhile, HSBC said it has completely overhauled its private banking business and has reduced the number of Swiss accounts by almost 70% since 2007.
In a statement, the bank said: "HSBC has implemented numerous initiatives designed to prevent its banking services being used to evade taxes or launder money."
The bank said it now puts compliance and tax transparency ahead of profitability.
But Panorama has spoken to a whistleblower who said there were still problems with tax dodging at HSBC private bank when she worked there in 2013.
Sue Shelley was the private bank's head of compliance in Luxembourg. She said HSBC did not keep its promise to change. "I think the verbal messages were great but they weren't put into practice and that disturbed me greatly," she said.
It was her job to make sure HSBC followed the rules, but she said she was sacked after raising concerns. She has since won a tribunal hearing for unfair dismissal.
Source: BBC