Malcolm Turnbull will soon release details of a parliamentary inquiry into MP citizenship, as he faces growing calls from his own ranks for an audit.
The inquiry comes as Stephen Parry formally resigned from the Senate, becoming the first Liberal to lose their seat over their dual citizenship in the latest scandal.
The prime minister also faced calls from Labor late on Thursday to reveal what he knew about advice reportedly given to Mr Parry by a minister in August that he should keep quiet about his eligibility doubts.
The ABC reported Mr Parry was told by the minister not to go public because the clear advice from the solicitor-general was that the court would clear coalition members.
Labor frontbencher Penny Wong said Mr Turnbull and Attorney-General George Brandis owed the public an explanation.
"This smacks of a cover-up and it's now time for some straight answers from Turnbull and Brandis," she told AAP on Thursday.
Senator Brandis told reporters he first heard about it when Mr Parry contacted him on Monday morning.
Immigration Minister Peter Dutton became the most senior coalition member to express interest in an audit, however, the prime minister and other cabinet ministers have hosed down the idea.
"An audit process sounds fine, but I just think there are holes all the way through it," Mr Dutton told 2GB radio.
It should not be the case that an obscure law in a foreign country can impact on the eligibility of an Australian to run for parliament, he said.
"The law does need to be modernised," Mr Dutton said.
Conservative Liberal backbencher Kevin Andrews said the issue required leadership from the prime minister.
"If I was the prime minister I would be requesting the Australian Electoral Commission to immediately undertake an examination of every MP and senator and report as soon as possible back to the government," the conservative MP told Sky News.
"If we don't do that, this is just going to continue to fester along and cause problems politically."
Education Minister Simon Birmingham described an audit as an unnecessary distraction and a waste of taxpayers' money.
"The prime minister is showing strong leadership, getting on with things that matter to the Australian people," he said.
Liberal backbencher Ted O'Brien told AAP an audit risks turning into a witch hunt and a better idea would be to seek advice from experts on a long-term solution.
"Let's address this with a calm and methodical approach so we solve it not only for the 45th parliament, but for the future parliaments of Australia," he said.
The joint standing committee on electoral matters is still awaiting the terms of reference for its inquiry.
A spokesman for the prime minister told AAP the terms were being drafted and would be available shortly.
Senator Wong said the only body that could determine the issue was the High Court not an audit.
The names of the replacements for four former senators will be made known by the High Court on November 10.
