Social Development Minister Bathabile Dlamini could be facing a motion of no-confidence in Parliament if Sassa can show no progress made in the grants handover by Tuesday.
The chairperson of the standing committee on public accounts, Themba Godi, said on Friday that he will push for a debate on Dlamini's fitness to hold office if a scheduled joint meeting next week bears no fruit.
A no-confidence vote in a Cabinet minister would be unprecedented, as it is the president's prerogative to hire and fire ministers.
Dlamini, Sassa and the South African Post Office are due before Scopa and the portfolio committee on Tuesday to account for their stalled deal.
- Read more: Sassa's Post Office deal in the balance
"If, on Tuesday, we don't get anything tangible, my personal view would be to request Parliament for an urgent debate on the matter, in which I will be pushing that Parliament should pass a vote of no-confidence in the minister," he told News24 reporters.
"If they come back on Tuesday and say there is no agreement, we will conclude that there is nothing substantive that they are bringing, and we will lay all of it at the door of social development."
If Dlamini fails to attend the meeting for the third time this month, Godi would also present the proposal to the committee.
A passed vote would not necessarily mean any action against Dlamini, as President Jacob Zuma has the final say over his Cabinet, but it would at least represent the will of Parliament.
Godi said that a deal with the Post Office to handle core banking functions in the grants system would give the struggling entity a major capitalisation boost.
Under the current leadership, which includes SAPO CEO Mark Barnes, the Post Office needs to be cleaned up and restored following "bad leadership decisions in the past", he said.
"We've always said that this very important service should be provided by a government entity. We do not think there should be any delay in all of this."
It would also take crucial grants money, contributed by South African taxpayers, out of the hands of private companies.
If the deal fails, Godi said, it would confirm suspicions that there was a concerted effort to have invalid service provider Cash Paymaster Services continue in the role.
"It will have achieved what some nefarious people wanted to achieve all along -- that CPS retain the contract beyond March. Which other company can take on that responsibility before March? We only have five months."
'Our core strength is Post Bank'
SAPO CEO Barnes said on Thursday that the current deal does not cater to the Post Office's core banking strengths.
Sassa's deal only included one of four possible services, excluded banking, cash payments to beneficiaries and production and distribution of Sassa cards.
"What they've asked us to do is develop a grants payments system, which is essentially an IT function," he told News24.
"We would argue that whoever develops that system needs to integrate it with their banking system. And the core strength the Post Office has is Post Bank."
It therefore does not make sense to ask the Post Office to only develop the IT system, and ignore three other core functions it tendered for.
Barnes did not want to pre-empt the process, as there were very important meetings to attend before Tuesday's showdown, nor would he be drawn on speculation surrounding reasons for the delay.
He did stress, though, that the time was getting "very, very tight".
"In any change of supplier, you need an extraordinary amount of co-operation between the incumbent and the newcomer. There are only so many hours in the day. But we will extend those hours for the national interest."
Post Office stability concerns
In a joint report to the Constitutional Court, however, Auditor General Kimi Makwetu flagged the Post Office's financial stability as a possible risk.
The report quoted the now-defunct Sassa "workstreams" concern that: "The vendor has provided information that is heavily reliant on a future state capability without proving current capability."
On Wednesday, Finance Minister Malusi Gigaba announced a R3.7bn bailout for the struggling entity.
Sassa is also concerned that the Post Office only has two thousand outlets, while the grants agency says it has ten thousand.
Sources close to the process say they are concerned that beneficiaries would spend more on travel costs to post offices in areas where they are not available.
The department did not respond to a request for comment by the requested deadline.
Sassa spokesperson Paseka Letsatsi told News24 that Dlamini will be holding a press conference on Monday morning in East London.
In August, disgruntled journalists staged a walk-out after Dlamini was two hours late for a media briefing on the grants issue in Polokwane.
Dlamini has failed to appear before two parliamentary committees over the lpast two weeks to discuss the deal, raising the ire of both ANC and opposition MPs.
MPs on the portfolio committee on social development said Sassa is heading for another crisis before the March deadline.