Temer told reporters he will not resign shortly after state-run news agency Agencia Brasil reported the news of the Supreme Court inquiry.
During a brief statement from Planalto presidential palace, Temer said he "never authorized payments to anyone to stay quiet."
The prominent daily newspaper O Globo reported
on Wednesday that a meat producer had recorded the President giving the go-ahead to bribe Cunha to "keep quiet" while he was in jail.
Temer's office released a statement denying that he had authorized any bribes to be paid to imprisoned former house speaker Eduardo Cunha in exchange for his silence regarding a long-running corruption investigation.
Temer, 76, said he will fight to prove his innocence.
"I know what I did, and I know my actions were right," he said. "I demand a full and quick investigation to clear up (the situation) for the Brazilian people."
According to the O Globo report, the information was revealed when the owners of the meat and chicken conglomerate JBS testified before the Supreme Court behind closed doors as part of a massive corruption investigation, dubbed "Operation Car Wash
," which implicates former and current politicians.
The corruption probe has led to the imprisonment of some of Brazil's most prominent politicians and business owners. More than 80 people have been charged with bribery and money laundering during Operation Car Wash.
Protests against unpopular President
Crowds gathered Thursday evening near Rio de Janeiro's Candelaria church, in the city center, carrying signs and flags demanding Temer's ouster.
Riot police surrounded the crowd, which filled several city blocks.
As night fell, some protesters threw Molotov cocktails at police, who fired tear gas into the crowd. There were no immediate reports of injuries.
Outside the Presidential palace, in Brasilia, dozens of people also gathered with signs and noisemakers accusing Temer of plotting a coup against former President Dilma Rousseff.
The latest political crisis comes as Temer experiences a sharp popularity drop. According to recent data released by pollster Datafolha, Temer's popularity rating is less than 9%.
Highest reaches of power implicated
was impeached last year
after the Senate found her guilty of breaking budgetary laws amid the swirling corruption investigation -- though she herself was not accused of corruption.
was Rousseff's former vice president and has been serving as interim President since her suspension in May 2016.
The wide-ranging corruption investigation began three years ago with the arrest of an executive at Petrobras, Brazil's government-run oil company.
Petrobras and Odebrecht
, Latin America's largest construction firm, played leading roles in the bribery ring.
The scandal is a central reason why Brazil is in its worst recession in history.
In addition to implicating Temer in the exchange with Cunha, Joesley Batista, the owner of JBS, reportedly testified that he had paid Cunha R$5 million since his imprisonment as part of an ongoing financial agreement, and still owed him R$20 million, in an agreement made after the politician exempted the poultry industry from certain taxes.
On Thursday, Batista issued a mea culpa on behalf of JBS, admitting that the company had no justifications for what it did.
"Our entrepreneurial spirit and immense will to work in the face of a Brazilian system that presented us with several difficulties to sell our products, led us to make the choice to make inappropriate payments to public agents," he said in a statement Thursday.
"Although we have explanations of why we did what we did, was have no justifications."
O Globo reported that Batista, his brother Wesley and five JBS employees testified as part of a plea bargain.
The statement from Temer's office acknowledges a meeting took place between Temer and Batista, adding "there was no dialogue that could compromise the President of the Republic's conduct."
The O Globo report also details other improprieties by Brazilian businessmen and politicians. As part of the investigation, "controlled actions" -- recorded conversations and exchanges of money which were tracked by federal police -- were carried out.
The testimony states that a congressman, Rodrigo Rocha Lourdes, was sent by Temer to "resolve a matter" concerning a JBS subsidiary. The lawmaker was later filmed receiving a bag containing R$500,000 ($159,000).
According to the report, Batista also testified that former finance minister Guido Mantega was his point of contact at the former ruling Worker's Party, the party of former leaders Lula and Rousseff, and said that Mantega would distribute bribes to Worker's Party members.
Rocha Lourdes' press secretary said he will "clarify all of the divulged facts" from the testimony.