Gwede Mantashe made the comment after deputy Finance Minister Mcebesi Jonas alleged that the Gupta family had offered him the minister’s post.
The family has denied the allegation.
The main opposition party is expected to ask Mr Zuma tough questions about the allegations in parliament.
Mr Mantashe is the third most powerful person in the governing African National Congress (ANC), and his comments suggest that Mr Zuma may be losing the confidence of influential members of the party, correspondents say.
Mr Zuma’s presidency has been marred by allegations of corruption, cronyism and incompetence, amid a worsening economic situation.
The crisis deepened on Wednesday, when Mr Jonas said that a member of the Gupta family had made “a mockery of our hard-earned democracy” by offering to promote him to the minister’s job last year.
In an interview with Bloomberg news agency, Mr Mantashe, the secretary-general of the governing African National Congress, said: “We need to deal with this; it will degenerate into a mafia state if this goes on.”
He is quoted by Reuters news agency as saying that no-one, including Mr Zuma, was “untouchable”.
The opposition has long accused Mr Zuma of letting the Guptas wield excessive influence.
The Guptas, who arrived in South Africa from India in 1993, have huge interests in computers, air travel, energy, and technology.
They said Mr Jonas’ statement was political point-scoring.
In 2013, there was an outcry after a private jet carrying guests to the wedding of a Gupta family member was allowed to land at a South African military air force base in Pretoria.
The opposition has said that links between President Zuma and the Guptas were so close that they have been nicknamed the “Zuptas”.
Source BBC