Labor believes cabinet minister Mitch Fifield's job will become untenable over what he knew about former Senate President Stephen Parry's dual citizenship.
The Senate referred Mr Parry to the High Court on Monday after it emerged he is a UK citizen, forcing his resignation from parliament last week.
The opposition supported the referral, but used a parliamentary debate to pursue Senator Fifield, who it says broke ministerial standards by not notifying the prime minister immediately after learning Mr Parry's citizenship status could be in question.
"It will become increasingly clear that Senator Fifield's position is untenable," Labor senator Don Farrell told parliament.
Senator Fifield denied he directed Mr Parry to keep doubts over his citizenship to himself, but repeated his admission he spoke with him in August before the High Court ruled against five MPs.
"I can't be definitive about when former Senator Parry started to reflect about his particular circumstances," Senator Fifield told parliament.
"While it would have been a couple of weeks before the High Court decision, it was not months."
He said he didn't tell other government members about the private conversation with Mr Parry.
Greens leader Richard Di Natale said it was important to get to the bottom of who knew what, and when.
"It appears for all intents and purposes that there has been cover-up here," Senator Di Natale said.
Earlier, independent senator Cory Bernardi hinted another senator could be holding dual citizenship.
He asserted the senator and government were aware of the possible breach of the Constitution but had yet to disclose the information.
"One of the senators knows they are not eligible to be here," he said.
But Attorney-General George Brandis rebuffed the claim, saying there was no evidence to support Senator Bernardi's assertion.