Later on Wednesday, the ANC postponed a meeting of its highest decision-making body, the National Executive Council (NEC), that had been scheduled for Wednesday in which Zuma's presidency was expected to be discussed.
The president, in power since 2009, is due to make a state of the nation address on Thursday, and some in the party want Mr Zuma to leave office ahead of that speech.
But Zuma loyalists say the president should complete his second and final term in office, which would end when elections are held next year.
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.
Cyril Ramaphosa released a statement clarifying the reasons for postponing the NEC meeting, which was due to take place on Wednesday. "We will be able to communicate further on President Zuma's position as president of the republic once we have finalised all pertinent matters". Developments are moving quickly with little explanation from parliament or the African National Congress.
Mr Ramaphosa has been cheered by many investors for his pledges to bolster growth, clamp down on graft and provide greater policy certainty.
"There is such huge expectation both internally and from the global community that people are going to have to be patient and understanding", Vandome said.
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As Fr Russell Pollitt, Director of the Jesuit Institute South Africa told Linda Bordoni, Zuma has repeatedly resisted calls to quit over corruption allegations.
The deputy president says he expects that he and Zuma will report to the ruling party and the country in the coming days.
Bongani Ngqulunga said a full cabinet meeting was scheduled for February 14, dismissing speculation in domestic media the embattled president had called a meeting to discuss his future with his cabinet colleagues.
Mbete said a new date for the state of the nation address would be announced shortly.
"It is not clear what the party will do", said Ebrahim Fakir, a political analyst, had said before the party's statement.
The party's NEC is split between supporters of Ramaphosa and Zuma.
In the meeting, Zuma allegedly shot down the two centre of power argument advanced by ANC Treasure General Paul Mashatile, reminding him how well the system worked in Gauteng, where Mashatile was chairman and David Makhura was provincial premier.
Zuma has been embroiled in scandals, including multi-million-dollar upgrades to his private home with state money and his association with the Guptas, a business family accused of looting state enterprises and influencing Cabinet ministers for their own benefit.