Peter Dunne has rejected suggestions from Australian government ministers of a conspiracy involving NZ Labour MP Chris Hipkins and an Australian Labor Party MP to bring down the deputy prime minister.

“This is so much utter nonsense – while Hipkins’ questions were inappropriate, they were not the instigator. Australian media inquiries were,” Dunne tweeted on Tuesday.

Labor leader Bill Shorten told caucus colleagues in Canberra – in a joke-laden speech in which he opened with the Maori greeting “kia ora” – the government was engaging in conspiracy theories and there was no need for MPs to engage in them.

“The Turnbull government is in a constitutional crisis caused by its own negligence,” Shorten said.

“The deputy prime minister, who I suppose we should now call the foreign minister or the leader of the ‘dual Nationals’ is unwittingly or not a dual citizen.”

However, Foreign Minister Julie Bishop says the contact between the Opposition and their cross-Tasman counterparts over the citizenship issue meant it would “very difficult to build trust” with a Labour government should the opposition party win the upcoming New Zealand election.

“I would find it very difficult to build trust with members of a political party that had been used by the Australian Labor Party to seek to undermine the Australian government,” Bishop said.

Cabinet minister Christopher Pyne accused Labor of stooping to new lows.

“Clearly the Labor Party are involved in a conspiracy using a foreign government, in this case New Zealand, the try and bring down the Australian government,” Pyne told Sky News.

Hipkins, in a parliamentary question last week, asked the NZ internal affairs minister whether a child born in Australia to a Kiwi father would automatically have citizenship. He’s yet to receive an answer.

NZ Labour leader Jacinda Ardern then pressed her MP why he had asked the question.

“He’s been very clear: Yes, someone from the ALP put some legal question to him around citizenship, no mention was made of anyone’s name, no rationale for any particular case being pursued was ever raised,” she told radio NZ.

“He asked the questions without knowing how that might be used and has made it very clear, in hindsight, had he known how it would be used, he would not have asked the questions.”

The High Court has been asked to rule whether Joyce’s NZ citizenship by descent made him ineligible for election to the Australian parliament.

Joyce has since applied to nullify his NZ status, which he would need to do if a by-election is held and he wants to recontest his seat of New England.

Prime Minister Malcolm Turnbull is also under pressure over comments he made in parliament about the likely outcome of the High Court ruling.

He argued the court “will so hold” Joyce’s right to sit in parliament.

Shorten said Turnbull should have learned from the “Yarra three” incident in which three Liberal MPs were forced to issue an unqualified apology to a Victorian court over their comments about “weak” terrorism sentencing.

The government has signalled it will refer the eligibility of five opposition MPs – including Justine Keay, Susan Lamb and Tony Zappia – to the High Court.

But the ALP says all caucus member are clear of constitutional problems.

– with AAP