Zuma could either be ousted by the ANC recalling him, or a vote of no confidence in parliament.
The African National Congress acknowledged growing calls for Zuma to step down, admitting to "serious and difficult disagreement" over the president's sacking of respected finance minister Pravin Gordhan last week.
Zuma could either be ousted by the ANC recalling him, or a vote of no confidence in parliament that has been scheduled for April 18.
The party retains a large majority in parliament, and Zuma has easily survived previous confidence votes.
ANC secretary-general Gwede Mantashe told reporters Wednesday that although the party had "reflected" on the resignation demands, "we won't recall President Jacob Zuma because opposition parties say so. It won't work that way."
Zuma has been under fire for several days -- including from within his own camp -- after dismissing Gordhan, a decision that caused the rand currency to plummet.
Gordhan's removal triggered unprecedented criticism from Deputy President Cyril Ramaphosa as well as the party's chief whip, its treasurer and several ANC allies.
Ramaphosa, who could succeed Zuma, described it as "totally unacceptable".
South Africa's powerful trade union federation Cosatu this week joined many anti-apartheid veterans, business leaders and civil action groups calling for Zuma to step down.
Mantashe -- who was among those who had hit out at Zuma in recent days -- blamed Gordhan's sacking on "the irretrievable breakdown" in relations with the president.
Gordhan was at loggerheads with Zuma for months, receiving support from several ministers and major foreign investors, as well as many ordinary South Africans.
"The (party) has emphasised the need for unity of the ANC and the alliance in the interests of South Africa," Mantashe said.
Gordhan had campaigned for budget discipline and against corruption, but Zuma's allies accused him of thwarting the president's desire to enact radical policies to tackle racial inequality.
Gordhan's sacking contributed to a credit ratings downgrade to junk status on Monday by Standard & Poor's, further fuelling calls for Zuma to step down.
The president has defended his change at the Treasury, saying that the government's financial policies remained the same.
S&P said the cut to below investment grade reflected "heightened political and institutional uncertainties" following the purge of Gordhan and other critical ministers.
With the cabinet overhaul exposing deep divisions within the ANC, the main opposition Democratic Alliance (DA) party hopes to benefit at general elections in 2019.
"No army anywhere in the world will subject their footsoldiers to be commanded by an enemy general," Mantashe said in relation to the no confidence vote.
Peter Attard Montalto, analyst at Nomura bank, said in a briefing note that Zuma was "a master tactician who can play the internal machinations of the ANC much better than anyone else."
The president has been accused of being in the sway of the wealthy Gupta business family, allegedly granting them influence over government appointments, contracts and state-owned businesses.
Since coming to power in 2009, Zuma has been hit by a series of corruption scandals, while the ANC suffered its worst ever results in local polls last year.
He is due to step down as head of the ANC in December, ahead of the 2019 general election.
The president is seen as favouring his ex-wife, former African Union chief Nkosazana Dlamini-Zuma, to succeed him -- rather than Ramaphosa.