The ANC has been dogged by infighting for much of this year as a series of corruption scandals have tarnished its image ahead of the December conference at which it will elect Zuma's successor.
The party is split between factions backing Nkosazana Dlamini-Zuma, a former minister and ex-wife of Zuma, and Deputy President Cyril Ramaphosa for the ANC's top job.
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ANC chief whip Jackson Mthembu told Reuters that whoever the party chooses next month, the incoming leadership should tell Zuma to go to allow the ANC to clean up its act.
"You can't keep him there," he said. Mthembu said the ANC could learn from what was happening in Zimbabwe, where the ruling ZANU-PF party is pushing for President Robert Mugabe to leave his post.
"In Zimbabwe they call that bloodless corrections ... We need to make the corrections immediately after the conference. How do you effect those corrections in government when the same person who might have contributed to a better degree still sits?" Mthembu asked.
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Mthembu is in the camp that backs Ramaphosa for ANC president and said it was important for the ANC to regain the trust of South African people after news reports that the Gupta brothers, business friends close to Zuma, had influenced government appointments and secured contracts from state firms. Both Zuma and the Guptas deny any wrongdoing.
Zuma's second term as president expires in 2019, but he could be forced out as head of state by the ANC's new leadership before his term ends, as was the case with former president Thabo Mbeki.
In May the ANC said its executive committee backed Zuma after calls for him to resign, and in August Zuma survived a no-confidence motion in parliament. Zuma still has strong support in the party, including from the influential women's and youth leagues as well as in rural areas, where several tribal chiefs back the traditionalist leader.
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The ANC has seen its electoral majority shrink over recent years, and some analysts predict it could lose the 2019 election.
Until recently that was unthinkable for a party that has led comfortably since sweeping to power under Nelson Mandela at the end of apartheid in 1994.
Mthembu said if the ANC failed to emerge from its December conference with a new image it was "doomed".
"It's us who got South Africa into this mess by electing Zuma to be president. We should have looked closely into the man. With hindsight we made a terrible error of judgment," he said.
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