South African President
Senior figures in the African National Congress met over the weekend after Zuma endured months of criticism over his sacking of respected finance minister Pravin Gordhan.
"There was a call made in the national executive committee for the president to consider stepping down," said ANC secretary general Gwede Mantashe.
"A number of members... were of the view that the ANC should listen to this call."
Mantashe said that the meeting had not backed the resignation demands, and had instead discussed "the need to reconnect with sectors of society that are drifting away from the movement."
A string of ANC allies have also urged Zuma to go, among them the Congress of South African Trade Unions (Cosatu), the South African Communist Party (SACP) and the South African Council of Churches (SACC).
Zuma has faced widespread public anger over a series of corruption scandals, record unemployment and a sluggish economy.
The crisis has seen two ratings agencies downgrade South Africa and brought tens of thousands of people onto the streets calling for Zuma's ousting.
The ANC is due to elect Zuma's successor as party leader in December, ahead of general elections in 2019.
The party -- which Nelson Mandela led to power in the 1994 post-apartheid elections -- has recently lost popularity, taking just 55 percent of the vote in last year's local elections, its worst ever result.
A campaign group of anti-apartheid veterans said that Zuma loyalists had "shown they clearly place their own narrow political and financial self-interests above... the best interests of the country."
Zuma retains support from ANC members in many rural areas and has been able to rely on party lawmakers to survive votes of no confidence in parliament.