Union general secretary Bheki Ntshalintshali said it was time for Zuma to "step down" after his purge last week of cabinet critics, which included the removal of the respected former finance minister, Pravin Gordhan.
Gordhan's sacking contributed to a credit ratings downgrade to junk status yesterday by Standard & Poor's as pressure on Zuma grew over the president's move to oust opponents within the cabinet.
"We no longer believe in his leadership abilities," Ntshalintshali said at a media briefing.
Cosatu, along with the South African Communist Party and the ANC, was at the forefront of the effort to dislodge white-minority rule in South Africa that led to non-racial elections in 1994.
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It has openly backed Zuma's deputy, Cyril Ramaphosa, who led Cosatu during the anti-apartheid struggle, to succeed him in 2019 when the president must stand down.
"Even if it means marching into the street we will do that to make our point. We believe in this alliance led by the ANC, but we want a reconfiguration of this alliance," said Cosatu president Sidumo Dlamini.
"(New Finance Minister Malusi Gigaba) will also be reaching out to the international investor community urgently to assure them," he said as he unveiled a South African-made railway locomotive in Pretoria.
"We are proud of having added many young ministers."
Cosatu's intervention came as Gigaba admitted that yesterday's downgrade to junk status was a blow to the economy.
"We acknowledge yesterday's announcement was a setback... but now is not a time for despondency," he told a media conference.
S&P said the cut to below investment grade reflected "heightened political and institutional uncertainties" following Zuma's shock purge of critical ministers.
The downgrade could mean that the government and state-run industries are forced to pay for their borrowing.
The shake-up has "put at risk fiscal and growth outcomes", the ratings agency added as it downgraded South Africa to BB+ from BBB-.
The rand fell three per cent against the US dollar following last night's downgrade. The rand was trading at 13.64 against the greenback at 1345 GMT today.
South Africa's leading financial paper Business Day called the downgrade "a black day for the country".
"It also specifically raised the possibility that the trust established between business leaders and labour representatives could be eroded," it wrote in its leading article.
Zuma's cabinet overhaul exposed deep divisions within the ANC, and officials from the main opposition Democratic Alliance (DA) party are hoping to recruit enough support from ruling-party MPs to unseat the president if there is a vote of no confidence.