The prime minister must explain to the country “exactly what’s been going on” with his family’s financial affairs in the wake of the Panama Papers leaks, and should be subjected to an investigation to determine whether tax has been avoided Jeremy Corbyn has said.

The Labour leader said he wanted HM Revenue and Customs to launch an investigation into all those implicated in the tax haven revelations, including David Cameron’s family.

Corbyn also argued that the government should consider imposing direct rule on British overseas territories and crown dependencies to stop them sheltering tax avoiders and evaders. Downing Street has insisted that the financial affairs of Cameron’s late father, Ian, which were detailed in the Panama leak, were a private matter.

Corbyn told reporters: “Well, it’s a private matter in so far as it’s a privately held interest. But it’s not a private matter if tax is not being paid. So an investigation must take place, an independent investigation, unprejudiced, to decide whether or not tax has been paid. “I think the prime minister, in his own interest, should tell us exactly what’s been going on.”

‘I have no family trusts’

Speaking after the launch of Labour’s local election manifesto, Corbyn said the HMRC investigation should look into “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”.

Asked whether the PM should resign if he is found to have benefited, the Labour leader said: “Let’s take one thing at a time. We need openness, we need an examination, we need a decision after that.”

Facebook Twitter Pinterest Jeremy Corbyn campaigning in Harlow. Photograph: Hannah Mckay/EPA

Pressed on whether he would publish his own tax return, Corbyn said: “There is no problem with my tax affairs. They are very, very limited indeed. I have got an income as an MP, sadly I have got no family trusts of any sort.”

Amid a global focus on the issue following a vast leak of tax records from a Panama law firm, Cameron is under scrutiny because the files revealed his late father ran an offshore fund that avoided having to pay tax in Britain. An aide to the prime minister said on Tuesday: “We’ve been clear, the prime minister doesn’t own shares in any company.”

He reiterated that questions relating to the Cameron family were private, and hit back at Corbyn by saying that Labour had failed to be tough on aggressive avoidance and evasion. “People have to judge governments on their record in government, and Labour’s was lamentable,” he said.

The aide gave the examples of thousands of the richest homebuyers not being charged stamp duty, and hedge funds easily cutting taxes by flipping the currency their accounts were in. He also said the government had asked for three things from the UK’s overseas territories and crown dependencies: the automatic exchange of tax information; a common reporting standard for multinational companies; and central registries so people know who really owns companies.

Facebook Twitter Pinterest Downing Street said the tax affairs of David Cameron and his father Ian were a ‘private matter’. Composite: Reuters

“They have all delivered on the first two, so now they must deliver on central registries. Gibraltar, Bermuda and Jersey have already done so. We are close to agreement with Guernsey and Isle of Man. Now we have to get the Cayman Islands and British Virgin Islands over the line and we are determined to work with them to do so by the time of the summit in May.”

Since the publication of the documents, the prime minister has come under fire for failing to live up to the tough words of his promises to crack down on tax avoidance and tax evasion and to bring transparency to offshore jurisdictions. Corbyn said the Panama Papers leaks had exposed an “industrial scale of tax evasion” and fleshed out his earlier proposal that the government could impose direct rule on administrations in places such as the Cayman Islands and British Virgin Islands to “chase down tax evaders”.

“I’m saying to those governments, hang on, what are you doing? You’re actually sheltering a large number of people who ought to be paying tax to the British government for the services for people from whom you’ve made the money,” he said. The Labour leader said overseas territories had been brought under direct rule during the financial crisis of 2008-09 and said it was simple to take them under control immediately.

‘Stop pussyfooting around’

He added that he hoped it would not be necessary to take this action. Instead, he said: “Let’s say bluntly to those governments, if you’re sheltering tax avoidance and tax evasion, stop now and pay your taxes.” Responding to comments from the former attorney general Dominic Grieve on Tuesday that shutting down tax havens in crown dependencies would just lead to them moving to territories beyond the UK’s control, the Labour leader said international action had to be taken but there was also a “moral imperative” for the wealthy to pay tax.

In his speech in Harlow, Corbyn said the Panama Papers “drives home what more and more people feel – that there is one rule for the rich and another for everyone else”. He added: “It is time to get tough on tax havens. Britain has a huge responsibility.

“Many of those tax havens are British overseas territories or crown dependencies. The government needs to stop pussyfooting around on tax dodging. There cannot be one set of tax rules for the wealthy elite and another for the rest of us. This unfairness and abuse must stop. No more lip service. The richest must pay their way.”

Reverberations from the leaked files have been felt worldwide, particularly in Iceland, Russia and Ukraine. France, Australia, New Zealand, Austria, India, the Netherlands and Norway are among countries to announce official investigations within hours of the publication of the leak, and many more could follow in the coming days and weeks.