The scandal erupted on Sunday when media groups began revealing the results of a year-long investigation into a trove of 11.5 million documents from the Panamanian law firm Mossack Fonseca, which specialises in creating offshore shell companies.
Among those named in the Panama Papers are close associates of Russian President Vladimir Putin, relatives of Chinese leader Xi Jinping and Iceland's Prime Minister Sigmundur David Gunnlaugsson, as well as Barcelona striker Lionel Messi.
Major international banks, including Credit Suisse, were also named in the leak among those that requested the most offshore companies for clients.
According to the papers, Credit Suisse created 1,105 offshore companies and UBS 1,100.
Offshore companies are not illegal and can be used for legitimate business needs, but commonly feature in corruption cases where they are used to secretly move ill-gotten gains abroad.
Credit Suisse CEO Tidjane Thiam said on Tuesday they would only endorse "legitimate" offshore arrangements.
"We only accept offshore structures, vehicles if they serve legitimate purposes," Thiam said, speaking in Hong Kong.
"Clearly, tax avoidance is not one of those," he added.
"We insist on knowing who is the beneficial owner. If it's not revealed, we will not engage in business with that entity," he added.
A UBS spokesman told AFP that the bank "conducts its business in full accordance with the applicable regulations".
"The bank has no interest in funds which are not in line with fiscal rules or which come from illegal activity," he added.
The Swiss banks' comments come after HSBC, another bank on the list of institutions that helped set up the most offshore accounts, also distanced itself from the revelations.
"The allegations are historical, in some cases dating back 20 years, predating our significant, well-publicised reforms implemented over the last few years," spokesman Gareth Hewett told AFP in an emailed statement.
The trove of Panama Papers documents was anonymously leaked to German daily Sueddeutsche Zeitung and shared with more than 100 media groups by the International Consortium of Investigative Journalists (ICIJ). More revelations are expected over the coming weeks.