The party’s chief whip in Parliament, Jackson Mthembu, said on Friday that voting against Mr. Zuma would be “tantamount to throwing a nuclear bomb” at South Africa, and that only a “bewitched” party would vote against its own president.
After Ms. Mbete’s ruling on Monday, a national spokesman for the A.N.C., Zizi Kodwa, said on Twitter that the party had “full confidence” that its members would vote to keep Mr. Zuma. And several small opposition parties, including the Communists, said they would vote against the measure.
Mr. Zuma has survived three no-confidence votes in Parliament. Another was amended into a vote of confidence, and then passed; yet another was withdrawn. He has also withstood an attempted impeachment motion, in 2016, and twice defeated votes challenging him as the party’s leader.
Leaked emails released in May, suggesting collusion between the Gupta family — which owns large companies in the technology, media and mining sectors — and senior A.N.C. members, has built pressure on Mr. Zuma as South Africa’s economy has slid into recession, its first since 2009.
The family’s relationship with the presidency has introduced a new portmanteau term into the South African political lexicon — “Zupta” — and prompted a critical report last year from the public protector, a national anti-corruption figure.
Somadoda Fikeni, a political analyst, said the no-confidence motion was unlikely to succeed, even in a secret ballot. “The A.N.C. doesn’t want to be seen changing its leadership on the back of pressure from the opposition, even though half the party believes that Mr. Zuma has become a serious political liability,” he said.
A more serious challenge to Mr. Zuma may come in December, when the A.N.C. is to hold its national conference and elect its next leader.
“The President’s opponents in the A.N.C. — and there are many — have clearly decided to wait until December to try and get rid of him,” said a political analyst, Steven Friedman. “The fight that matters is taking place within the party. Until we have millions of people marching on the streets, instead of merely thousands, the A.N.C. is going to remained focused on this internal struggle.”
Mr. Friedman cautioned in an opinion essay last week that the secret ballot could set a damaging precedent.
Protesters gathered in Cape Town on Monday afternoon, led by a coalition of civil society and religious leaders. A brief but spirited counterprotest by members of the A.N.C.’s Umkhonto We Sizwe Military Veterans Association, numbering some 50 people, was disbanded by police.
The marchers were addressed outside Parliament by Mcebisi Jonas, a former deputy finance minister axed by Mr. Zuma in a contentious cabinet reshuffle in March.
The protest, organized by a coalition of more than 20 civil society organizations, was a “call for accountability from A.N.C. members,” said its organizer Mandisa Dyantyi, deputy secretary general of the Social Justice Coalition, which helped organize the protest. “They’re in Parliament to represent the people who elected them, not their party,” she said.
Mr. Jonas — who has accused a member of the Gupta family of offering him a bribe, a claim the family has strongly denied — told the protesters: “We cannot allow our freedom to be sold so cheaply. We have to fight. We’re gathered here to shape the future of this country.”
“This march is a distraction from the real issues in South Africa,” said Banzi Siwe, 23, a student, who added: “Compared to what the white people stole before, this is nothing.”
Two further protests are planned in Cape Town for Tuesday, one led by opposition parties and one in support of Mr. Zuma by the A.N.C. branch for the Western Cape metropolitan area. This has spurred fears of clashes between rivals. “The likelihood of this turning into violence is our biggest concern,” said Nomfundo Mogapi from the Center for the Study of Violence and Reconciliation, a local think tank.
Khaya Yozi, a spokesman for the A.N.C. in Cape Town, said that the party’s members “would not be provoked” by opposition protesters. “We need to set aside the divisions within our party and protect the A.N.C.”
An earlier version of this article referred incorrectly to a spokesman for the African National Congress, Zizi Kodwa. He is a man, not a woman.
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