Labour has demanded an independent investigation into the tax affairs of those implicated by the records, which includes details about the Prime Minister’s dead father, Ian Cameron.
During a visit to Birmingham, Mr Cameron said the information should be used to ensure that companies and individuals are paying the tax they should but sidestepped calls for a probe.
He said: “In terms of my own financial affairs, I own no shares. I have a salary as Prime Minister and I have some savings, which I get some interest from and I have a house, which we used to live in, which we now let out while we are living in Downing Street and that’s all I have.
“I own no shares, no offshore trusts, no offshore funds, nothing like that. And, so that, I think, is a very clear description.”
Mr Cameron declined to say if his family had benefited in the past or were likely to in the future from the offshore arrangements highlighted in the papers.
He insisted that “no Prime Minister has done more” to crack down on tax evasion and aggressive tax avoidance.
Mr Cameron’s father ran an offshore fund which avoided ever having to pay tax in Britain by hiring a small army of Bahamas residents - including a part-time bishop - to sign its paperwork, according to the Guardian.
Ian Cameron, who died in 2010, was a director of Blairmore Holdings Inc, which, until 2006, used unregistered “bearer shares” to protect its clients’ privacy.
His use of the firm to help shield investments from UK tax helped build up a significant legacy, part of which was inherited by the Prime Minister.
There is no suggestion that this avoidance arrangement or others exposed by the leak were anything but entirely legal or that Mr Cameron’s family did not pay the UK tax due on any repatriated assets.
Jeremy Corbyn dismissed suggestions by Downing Street that the family’s tax arrangements were a “private” matter.
The Labour leader said: “It’s a private matter in so far as it’s a privately held interest, but it’s not a private matter if tax has not been paid.
“So an investigation must take place, an independent investigation.”
Mr Corbyn said he wanted an investigation conducted by HM Revenue and Customs “about the amount of money of all people that have invested in these shell companies or put money into tax havens and to calculate what tax they should have paid over the years”.
The Labour leader also suggested the Government could intervene to take direct control of the UK’s offshore tax havens.
“If it’s necessary for ministers to intervene because the governments of the Overseas Territories won’t act, they can use an order in council to take control of them immediately,” he said.
“The Prime Minister has championed the transparency agenda at a series of international summits, and legislation forcing British companies to disclose who owns and benefits from their activities comes into force in June.
“But despite several years of pressure, few UK Crown Dependencies and Overseas Territories - said to make up a large part of the tax havens referred to in the papers - have taken concrete action to open up the books.”