National and Act say that Parliament should return next week in some form, despite the level 4 lockdown currently scheduled to run until Tuesday night.
But the Government has shot back, saying that National's message in public is different to what it argued in the privacy of the committee - an allegation National disputes.
Leader of the House, Chris Hipkins' office said National actually agrees Parliament should not meet, but wants the Government to call it off on its own, rather than seeking the agreement of other political parties.
National leader Judith Collins earlier said she would like Parliament to return.
"We are pushing for Parliament to come back in some form. We are very aware that other countries around the world have been able to bring back their parliaments and operate through lockdowns.
"We are an essential service - particularly in a democracy," Collins said.
Act leader David Seymour agreed Parliament should return.
"The AM show and the Project can operate socially distanced, surely Labour don't want to shut down a democratic Parliament.
"The supermarkets are operating, food is important, but so is democracy," Seymour said.
"If they are not prepared to open Parliament, then they should have an Epidemic Response Committee like last lockdown.
"This country has faced crises before, we've never abandoned democracy,"
Parliament adjourned during last year's level 4 lockdown, but a separate committee to scrutinise the Covid response was set up in place of the House sitting.
Parliament resumed in social distanced form under level 3.
This week Parliament was in recess, meaning it did not sit, however, it was scheduled to resume next Tuesday.
The decision on whether Parliament should return will be made by Parliament's Business Committee, which includes members from all parties.
It met this afternoon, but did not reach a decision.
The Government has the upper hand. Not only does it have a majority on the business committee, but Parliament's rules, the standing orders, allow the Government to postpone sitting during an epidemic.
Standing Order 55 allows the Prime Minister to write to the Speaker of the House requesting Parliament's sitting be pushed back for a month.
But she can only do this if there is an epidemic notice in force (which is currently the case) and after receiving a written recommendation from the Director-general of Health, Ashley Bloomfield, that Parliament should be postponed.
She would also have to consult with other party leaders.
If Parliament were to be postponed for more than one month, all party leaders would have to agree.
The Business Committee could also agree to postpone Parliament, but this would require near unanimity on the committee.
Politically, it would also allow the Government to show that it was not acting unilaterally, but with the support of the other parties in Parliament.
The Prime Minister acting unilaterally will potentially make the Government look bad, while using the Business Committee will give the proceedings the imprimatur of unanimity.
The proceedings of the committee are secret and subject to privilege.
The looming question is whether National and Act put forward a counter-proposal for allowing Parliament to resume in a safe way under level 4, or whether they cornered the Government into acting unilaterally by using standing order 55.
A spokesperson for Hipkins, who is the Government's representative on the committee, said National agrees Parliament should not meet but thinks the best way to stop it meeting is by triggering standing order 55, not using the business committee.
But National's shadow leader of the house Chris Bishop said Hipkins' office "completely misrepresented" National's position.
"As per the comments by Judith Collins, National would prefer Parliament to meet next week. The Government's position, as we understand it, is it can't, therefore the Prime Minister needs to take the appropriate actions under the standing orders for it not to meet,"
There is also a dispute over whether an Epidemic Response Committee could be reestablished.
Parliament would need to meet again to reestablish the committee, which would somewhat defeat the purpose of it being established as an alternative to Parliament.
But a counter-proposal was to repurpose the already existing Governance and Administration Committee, which National chairs, as the new Epidemic Response Committee.