Labor has dismissed Prime Minister Malcolm Turnbull’s calls for an end to the citizenship “witch hunt”, signalling it will go on the attack unless the government agrees to have all politicians produce documents to prove their eligibility.

The opposition on Sunday directly named two lower house government MPs that should be referred to the High Court – that is unless Mr Turnbull heeds a call for all politicians to disclose documents that prove their eligibility to serve in Parliament.

That warning came hours after Assistant Minister for Immigration Alex Hawke was on Sunday forced to deny claims he holds Greek citizenship through his mother.

“I am an Australian citizen only and have never held or acquired or sought Greek or any other citizenship,” Mr Hawke told News Corp.

But Labor’s leader in the Senate Penny Wong said on Sunday that “the status quo is not acceptable”.

“Turnbull either accepts Labor’s proposal for universal disclosure or he refers MPs in doubt – including Alex Hawke and Julia Banks – to the High Court,” she said.

Ms Banks has also faced suggestions she inherited Greek citizenship, claims she has denied.

The opposition’s proposal, which would ask politicians to demonstrate to the Parliament how they are not dual citizens, has been dismissed by the government as it warns of a “witch hunt”.

On Friday, Energy Minister Josh Frydenberg described reports he may be a Hungarian citizen as “absurd”, sparking a fiery defence from the Prime Minister and claims of a “witch hunt”.

No need for a ‘witch hunt’, says PM

Government ministers pushed back strongly against Labor’s proposal for a wider audit on Sunday, saying that parliamentarians could already move to refer MPs and senators to the High Court.

“So no witch hunt, no Salem witch trial will actually take away from the fact that the High Court is the only arbiter on this particular question,” Financial Services Minister Kelly O’Dwyer told the ABC.

“So I think we need to take a pretty deep breath, because those people who said they’re not compliant have, in fact, come forward and the High Court has made a determination and it is clear.”

With the government short of a majority while Barnaby Joyce fights a byelection in New England, the opposition has ramped up it assault over citizenship.

Opposition Leader Bill Shorten backflipped to support the universal disclosure proposal on Friday, but Labor has stopped short of backing an audit of politicians’ eligibility, as backed by the crossbench and a growing number of government backbenchers.

When the Senate sits next on November 13, the Greens will move to establish a Senate committee that would ask citizenship experts to assess the eligibility of parliamentarians.

“We have to start to restore some integrity in the nation’s parliament,” Greens Leader Richard Di Natale told the ABC.

Some government MPs have backed an audit, while others have hit back at Labor, demanding that in-doubt opposition politicians such as Tasmania’s Justine Keay also prove their status.

Liberal MP Sarah Henderson has demanded Labor’s Justine Keay, who has faced questions over possible British citizenship, produce her renunciation papers.