SÃO PAULO—Allegations by executives at meat processor JBS SA that they paid bribes to the president and his two predecessors mark an escalation of the clash between Brazil’s political establishment and its business magnates, putting the company and the government at further risk.

JBS’s allegations, leaked by Brazil’s O Globo newspaper on Wednesday and released by the Supreme Court on Friday, come a month after the court ordered investigations into one-third of sitting ministers following disclosures by construction giant Odebrecht...