A court ruling, handed down on September 12, cleared Zuma of corruption charges, alleged that Mbeki's government had interfered in the decision to prosecute him, which Mbeki's cabinet has denied.
After the ruling, there was intense speculation over whether the ANC would ask Mbeki to quit, push him out in a vote of no confidence or allow him to serve out his term of office, which is due to end next year.
The ANC appeared unable to reach a decision on the issue at a meeting on Friday and canceled a press conference where officials had been scheduled to announce their decision.
"The debate is actually in the middle of nowhere. Everybody is expressing their view and then when we complete that debate and take a decision we will come and communicate with you," Gwede Mantashe, the ANC secretary general, said.
Following the ruling in Zuma's trial, the ANC youth league called for Mbeki to leave but the ANC as a whole reacted more cautiously, saying any decision would be by consensus and announced after its national executive committee had met over the weekend.
Half the cabinet is reportedly threatening to walk out if Mbeki is forced to leave.
The president issued a statement on Friday saying: "It impoverishes our society that some resort to the tactic of advancing allegations with no fact to support these.
"The question will have to be answered now - what kind of society are we building, informed by what value system and with what long-term effect to the political and overall moral health of the nation?"
Mbeki took over as South Africa's president in 1999.
The dismissal of corruption charges against Zuma makes it possible for him to become South Africa's president after next year's election.