Scattered protests sprang up in front of the presidential palace and along Sao Paulo's main avenue as opposition lawmakers and even a high-profile ally called for Temer to step down.
"Given the gravity of the situation and the responsibility to keep Brazil from plunging into the imponderable, the only option is for President Michel Temer to resign," said Senator Ronaldo Caiado, leader of the government-allied Democratas party in the Senate.
JBS, the world's biggest meatpacker, declined to comment. A senior JBS executive said there had been no formal communication about the matter within the company.
While the O Globo report described an unprecedented sting operation mounted by Brazil's federal police in conjunction with senior JBS executives to snare several politicians, Temer was the one target that towered over the rest.
Batista used a hidden device to record an alleged discussion with Temer about hush money the executive was paying to Cunha, according to the newspaper.
The report did not say what Cunha was asked to keep quiet about. When Batista told Temer he was paying Cunha to remain silent, the president was recorded saying, "You need to keep that up, okay?" according to the newspaper, which did not say how it had obtained the information.
Cunha, once a powerful member of Temer's ruling party, has previously said he had compromising information about several senior politicians linked to a vast political bribery scandal at state oil firm Petroleo Brasileiro SA, or Petrobras.
O Globo reported that Batista and his brother, JBS Chief Executive Wesley Batista, presented the recording to prosecutors as part of plea bargain negotiations underway since March.
Temer's office said in a prepared statement that he "did not participate in or authorize any activity with the aim of avoiding a plea bargain or collaboration" by Cunha, adding that the president supported a full investigation of the allegations.
Brazil assets tank
An exchange-traded fund tracking Brazil's benchmark Bovespa stock index fell more than 8 percent in Tokyo, while American depositary receipts for Petrobras fell as much as 11 percent in after-hours New York trading as investors ditched Brazilian assets at the sight of more political turmoil.
Despite his extreme unpopularity among Brazilian voters, Temer has maintained a broad majority in Congress, but analysts said public outrage over the scandal could change the political equation.
"If the recordings are confirmed, Temer would have to defend his mandate on multiple fronts," said political analyst Thomaz Favaro of the consultancy Control Risks, citing the danger of public pressure for Temer's resignation, lawmakers' calls for impeachment and a possible investigation by the nation's top prosecutor.
"The development thus substantially increases the risk of an unscheduled government change in Brazil before the 2018 general elections," Favaro told clients in a note.