The admission by deputy finance minister Mcebisi Jonas that he was offered the top job in the treasury by the Gupta family caused outrage in a country already alarmed by a series of corruption scandals.
The Gupta brothers Ajay, Atul and Rajesh were smalltime businessmen back home but have built up a string of companies with interests in computers, mining, media and engineering since moving to South Africa in the 1990s.
They have long been accused of wielding undue influence over Zuma, whose son Duduzane is a partner in some of their businesses. Zuma's third wife also used to work for them.
The newspaper quoted the message from an unnamed businessman as reading: "Please keep your own counsel. Martyrdom is best left to Christ."
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In his statement yesterday, Jonas said: "Members of the Gupta family offered me the position of Minister of Finance to replace then-minister Nene. I rejected this out of hand."
The Guptas, who had attracted controversy a few years ago by flying in wedding guests at the Waterkloof air base which is normally reserved for visiting heads of state and diplomatic delegations, issued a statement denying Jonas's claims.
"We need to deal with this; it will degenerate into a mafia state if this goes on," ANC Secretary-General Gwede Mantashe told Bloomberg News.
"The fact we are talking about this so boldly now shows that things are going to change."
The ANC, which led the struggle to end apartheid, holds a meeting of its national executive committee starting tomorrow, where the Gupta issue is likely to dominate.
The official opposition party in parliament, the Democratic Alliance (DA) said the latest developments make Zuma's position "increasingly untenable".
The party said it would lay corruption charges against the Gupta family.