A special meeting of the ANC's National Executive Committee on Wednesday was postponed yesterday to allow Zuma and Ramaphosa to settle discussions and report back to the ANC and the country in the coming days.
South Africa's President Jacob Zuma told African National Congress (ANC) leaders the governing party will lose the 2019 elections if he steps down, said Economic Freedom Fighters (EFF) leader Julius Malema.
Earlier, parliament made a decision to ask Zuma to delay his state-of-the-nation address due to fears of violence, National Assembly Speaker Baleka Mbete told reporters outside parliament in Cape Town.
When pro-Zuma forces in the party humiliated former President Thabo Mbeki by forcing him out of office in 2008, eight months before the end of his term, it precipitated a split in the party.
Mr Zuma had "abused the trust" of South Africans and "betrayed the country Nelson Mandela dreamed of", the foundation, which is run by close colleagues of South Africa's first black president, said.
On Wednesday the ANC's National Executive Committee will meet.
In a separate case, Zuma's lawyers last week submitted arguments to prosecutors about why he shouldn't be prosecuted for corruption charges tied to an arms deal two decades ago.
Ramaphosa said he understood that the uncertainty circling Zuma's future would be a cause for concern for many South Africans, but remained confident that the pair could reach an outcome that would unite the country.
The meeting between the two came just hours after Parliament announced the postponement of the state of the nation address - the first to happen since 1994 -giving no alternative date.
The power struggle has rocked the ANC, the much-celebrated liberation party which led the fight against white-minority rule but has since lost much of its public support. A statement from the presidency later claimed Mr Zuma had requested the delay.
"We wanted him to go a long time ago", Sizwe Pamla, the federation's spokesman, said by phone from Johannesburg.
The president has resisted calls to quit over corruption allegations.
"This is the beginning of the end for Zuma", said Theo Venter, a political analyst at North-West University's business school in Potchefstroom, west of Johannesburg.
The 80-member ANC committee could "recall" Mr Zuma from office - an instruction he could constitutionally refuse to obey, triggering political chaos. There is very little likelihood of uneventful State of the Nation Address on Thursday, The Citizen reported.
Zuma's defiance of calls to resign has stolen some of the shine from the optimism generated by the victory of Ramaphosa, who's cheered by many investors for his pledges to bolster growth, clamp down on graft and provide greater policy certainty.
While the legislature is due to debate a motion of no-confidence proposed by the opposition Economic Freedom Fighters party on February 22, it "may not be needed", the ANC's chief parliamentary whip, Jackson Mthembu, told eNCA television.
"The king is a bargainer, he could help Cyril (Ramaphosa) heal KwaZulu-Natal after Zuma goes".
He has admitted that there has been serious corruption within the government and pledged to clean up state companies such as debt-laden power monopoly Eskom. At this stage President Zuma is still scheduled to deliver the address.