A senior South African government official said Wednesday that a wealthy family close to President Jacob G. Zuma offered him a cabinet position, a revelation that added to mounting corruption allegations against the president. The official, Deputy Finance Minister Mcebisi Jonas, said that members of the Gupta family, which has extensive business interests in South Africa, asked him to replace Nhlanhla Nene, who was dismissed in December. Mr. Jonas said he rejected the offer from the Guptas. “The basis of my rejection of their offer is that it makes a mockery of our hard-earned democracy, the trust of our people and no one apart from the president of the republic appoints ministers,” Mr. Jonas said in a statement. The family denied the deputy minister’s claim. Mr. Zuma is scheduled to answer questions from lawmakers on Thursday, where opposition parties say they will seek clarity on the political influence of the Gupta family.