"I know what I have done", he said, according to a BBC translation.
On Thursday, the Supreme Federal Tribunal, the country's highest court, suspended Neves from office indefinitely.
"If Temer doesn't fall, he will lead a walking dead administration". Cunha was later imprisoned on a 15-year sentence for corruption. The paper says that this tape is in the hands of prosecutors who have, for the last few years, been carrying out the so-called vehicle wash investigation.
Temer said he would not resign.
The news sparked calls for Temer's impeachment, halted debate on controversial neoliberal reforms and raised serious questions about the ability of the government to survive until the 2018 presidential election.
The scandal has engulfed Brazilian politics, with a third of Mr Temer's cabinet under investigation for alleged corruption. Temer is denying a report that he endorsed the alleged bribing.
A newspaper reported that Temer was caught on tape discussing hush money for jailed former speaker of the house Eduardo Cunha.
A presidential aide quoted by Reuters said Temer still believed that a recording purportedly between business executive Joesley Batista and himself contained nothing incriminating.
Temer's office issued a statement saying: "President Michel Temer never solicited payments to obtain the silence of former deputy Eduardo Cunha".
Since he became president a year ago following the controversial impeachment of Rousseff - who was removed from office accused of manipulating federal budgets, his government has been plagued by scandal and several ministers have been forced to resign.
There were immediate demands late Wednesday from leftist opponents for Temer's removal. Opposition politicians called for his impeachment.More news: Senator McCain Calls for Expulsion of Turkish Ambassador
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Demonstrators march carrying a banner that reads in Portuguese "Get out Temer and your reforms" in Sao Paulo on Thursday.
He went on to warn that the revelation of the "secretly recorded conversation" threatened to intensify the country's political crisis to an unprecedented level, rendering the "immense effort to pull the country out of recession" useless.
Federal police, meanwhile, intensified their investigations in the corruption probe, dubbed 'Operation Car Wash, ' and closed in on Temer allies.
In 2016, Temer and his associates ousted the elected president Dilma Roussef, accusing her of manipulating statistics to hide the country's economic woes. Investors were panicked by allegations of witness tampering involving the president.
But the newest scandal around Temer appears to be more destabilising, raising the possibility of even greater turmoil. "It is necessary that Brazil find a way through direct election", one protester, Thiago Pereira, told RT's Ruptly video news agency.
Petrobras has been at the centre of the political scandal in which dozens of politicians, officials and businesspeople are accused of having been part of a pay-to-play system focused around the company.
Apparently the brothers reached an agreement with prosecutors which involved helping pin down Temer, and powerful coalition leader Aécio Neves, as part of a plea bargain.
For Jose Carlos Martins, head of construction industry group CBIC, political paralysis would further undermine an economy struggling with more than 14 million unemployed.
In his speech, Temer categorically denied the allegation and insisted he will stay on the job.
It is not yet clear why Temer would want to buy Cunha's silence, since the tape itself will not be made public until the Brazilian Supreme Court approves its release. Temer then allegedly told Batista: "You need to keep that up, okay?"