ANC party officials met with Zuma on Sunday to request his resignation, but he refused, triggering a week of dramatic political manoeuvering.
Cyril Ramaphosa, the president-in-waiting, has said that "transition" talks with Zuma were making progress.
But he gave no further details and admitted that the process was causing "uncertainty surrounding the position of head of state".
Leader of the main opposition Democratic Alliance (DA) party Mmusi Maimane warned that South Africa could not "be held in limbo while Cyril Ramaphosa and Jacob Zuma... fight over the terms of Zuma's exit".
Zuma, whose rule since 2009 has been dominated by graft scandals, has been under growing pressure to quit in favour of Ramaphosa, his deputy and the new African National Congress (ANC) party leader.
Maimane also said that Zuma, who is facing numerous legal cases, must not be offered criminal amnesty as part of a resignation deal.
"He is not above the law," Maimane said, predicting that "Jacob Zuma will retire in prison."
The TimesLIVE news site today published audio of ANC treasurer general Paul Mashatile saying that Zuma had refused to resign and that the ANC had been ready to recall him at a crunch meeting.
That gathering, scheduled for yesterday, was cancelled at short notice with Ramaphosa saying the move would "enable President Zuma and myself to conclude our discussions... in the coming days".
"Normally when a sitting president is asked to step down, it happens. But President Zuma has declared his unwillingness," said Amanda Gouws, a political science professor at Stellenbosch University.
The power struggle has rocked the ANC, the renowned liberation movement that under Nelson Mandela led the fight against white-minority rule but has since lost much public support.
Despite the attacks on him, Zuma still enjoys some backing within the party, particularly among rural members and within his own Zulu community.
His presidency has been marked by an economic slowdown, record unemployment and multiple allegations of corruption.
He faces several court cases, including action relating to 783 payments he received allegedly linked to an arms deal before he came to power in 2009.
Save South Africa, an anti-Zuma campaign group, said it was "deeply concerned at the veil of secrecy surrounding negotiations" over Zuma's exit.
"Shuffling Zuma out of the back-door, with any number of trade-offs and compromises, is not only bad for Mr Ramaphosa - it is bad for South Africa," it said.
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