The Federal Opposition is softening its stance on an audit of all parliamentarians' citizenship, but still needs to be convinced how such a process would work.

Six politicians have lost their jobs in Canberra's dual citizenship fiasco, with the departure of former Senate president Stephen Parry resulting in a surge of public support from the Coalition's backbench for an investigation of members and senators' nationality to be launched.

Until now, the Turnbull Government and the Labor Party have been advocating against an audit.

But the ABC understands Opposition Leader Bill Shorten could still back the proposal and senior Labor figures are of the opinion the party has everything to win and nothing to lose from such a process.

Despite the party's staunch defence of its own vetting processes, the view is that Labor would remain in Opposition if any of its members were knocked out in an audit.

In the event the Coalition suffers casualties, it could prompt by-elections and threaten Malcolm Turnbull's Government.

A senior Labor source suggests Mr Shorten remains unconvinced about exactly how such an audit would work, and has concerns about any legal and constitutional issues it could cause.

Labor's Senate leader Penny Wong argued nothing would replace the authority of the High Court in adjudicating such cases.

Mr Parry's resignation, just four days after the High Court ruled five other politicians had breached the constitution's ban on dual citizens sitting in Parliament, has prompted Liberal backbenchers to join the demands for an audit.

"This is now a festering sore for the Parliament," former defence minister Kevin Andrews told AM.

"If it's not dealt with now, it'll just continue to be an issue, there'll be investigations into other people, it'll be an ongoing issue."

Tasmanian Liberal senator Eric Abetz argued his former colleague Mr Parry did the right thing in owning up to his dual citizenship, but said the public's faith in their elected officials was slipping.

"I have never been one to suggest that all wisdom resides between my two ears, but I believe that this is a matter that needs to be resolved for the integrity of the Parliament, to enable Australians to retain confidence in the Parliament," he told Lateline.

"In those circumstances I think some sort of check or an audit would be of benefit for the body politic of Australia."

Turnbull should say who knew what and when: Wong

The Federal Opposition has seized upon reports members of the Coalition knew about Mr Parry's potential dual citizenship months before he publicly revealed his concerns.

Former Senate President Stephen Parry's departure from Parliament could be the trigger for a citizenship audit ( AAP: Mick Tsikas )

The ABC revealed Mr Parry confided in senior colleagues including Cabinet minister Mitch Fifield months ago, but a colleague advised him not to raise the alarm because the Government had advice senators like Fiona Nash would be allowed to remain in Parliament.

Senator Wong said the saga was starting to "smack of a cover-up".

"Both Malcolm Turnbull and George Brandis have said they knew nothing about this — they told Australians they were completely shocked," she told AM.

"Well, now we've learned that at least one Cabinet minister did know, not days ago but months ago.

"The Prime Minister really needs to step up, find out who knew, when they knew, and why his ministers covered it up for so long."

Senator Fifield confirmed Mr Parry had approached him a few weeks ago with his concerns, and he had encouraged him to investigate his family history.

Mr Parry is understood to have felt betrayed by senior members of the Government who have criticised him for waiting until after the High Court's decision to reveal his predicament.