The AFR says pressure was now building on New Zealand to "tighten its tax loophole for foreign trusts."
It says: "New Zealand's 12,000-plus offshore trusts pay no New Zealand tax on foreign earnings. Their beneficiaries are not registered and their accounts are not filed with any public body. New Zealand regulators may demand this information, but it is not disclosed to foreign governments."
• INTERACTIVE: How multinationals pay no tax in NZ
AFR says that the Panamanian law firm has strong links to New Zealand - with trusts named in its probe including those named after New Zealand locations - such as Rotorua and Haast.
Gerard Ryle, director of the International Consortium of Investigative Journalists, said the documents leaked - which were passed to German newspaper Suddeutsche Zeitung - covered the day-to-day business at Mossack Fonseca over the past 40 years. Some 107 media organisations in 78 countries have been analysing the documents.
"I think the leak will prove to be probably the biggest blow the offshore world has ever taken because of the extent of the documents," said Ryle.
• The data contains secret offshore companies linked to the families and associates of Egypt's former president Hosni Mubarak, Libya's former leader Muammar Gaddafi and Syria's president Bashar al-Assad.
• It also reveals a suspected billion-dollar money laundering ring that was run by a Russian bank and involved close associates of President Putin.
Mossack Fonseca told the BBC it had always complied with international protocols to ensure the companies it incorporated were not used for tax evasion, money-laundering, terrorist finance or other illicit purposes.
The company said it conducted thorough due diligence and regrets any misuse of its services.
"For 40 years Mossack Fonseca has operated beyond reproach in our home country and in other jurisdictions where we have operations. Our firm has never been accused or charged in connection with criminal wrongdoing.
"If we detect suspicious activity or misconduct, we are quick to report it to the authorities. Similarly, when authorities approach us with evidence of possible misconduct, we always cooperate fully with them."
Mossack Fonseca said offshore companies were available worldwide and were used for a variety of legitimate purposes.