He has survived several parliamentary votes but this is the first time the ANC will formally discuss his position.
The top ANC body has extended its meeting to debate the motion, tabled by Tourism Minister Derek Hanekom.
Mr Zuma has increasingly come under criticism over corruption allegations, with a recent report highlighting his links with the wealthy Gupta family.
At least two other cabinet ministers are understood to be backing Mr Hanekom’s proposal – Health Minister Aaron Motsoaledi and Public Works Minister Thulas Nxesi.
The BBC’s Karen Allen in Johannesburg says this could shape into the toughest battle of Zuma’s career.
Earlier this month, an investigation by the country’s anti-corruption watchdog said a judicial inquiry should be set up to further investigate allegations of criminal activity in Mr Zuma’s government.
The investigation found evidence that the Guptas, a business family with links to Mr Zuma, may have wielded undue political influence over the appointment of ministers.
Both Mr Zuma and the Guptas have denied any wrongdoing.