Temer took over previous year after the impeachment of president Dilma Rousseff, a political quake to a large extent engineered by the then-powerful Cunha.
President Temer confirms meeting with one of the founders of slaughterhouse JBS but denies having bought Cunha's silence.
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. The company is caught up in Brazil's political bribery scandal and is accused of paying kickbacks to Eduardo Cunha, a former Speaker of the lower house of parliament, who is in jail.
Globo also reported that Temer was recorded by Batista endorsing the bribing of a former lawmaker to keep him quiet.
In a statement late on Wednesday, the president's office said Mr Temer did not participate or authorise any attempt to keep Cunha from reaching a plea bargain with justice officials. Temer allies, however, argue the Amazonas case is different and insist only Rousseff should be punished.
President Temer is already losing support in Congress. Many Brazilians expressed shock on social media when Temer finally spoke Thursday night and said he would stay in power. The top court suspended Neves from the Senate.
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Temer told reporters he will not resign shortly after state-run news agency Agencia Brasil reported the news of the Supreme Court inquiry.
Over the past several years, dozens of former executives and politicians have been arrested in the graft investigation known as Car Wash that has exposed billions in bribes and tarnished almost every prominent member of Brazil's establishment.
Temer quickly set about introducing market reforms to try to get Brazil's floundering economy back on the rails.
The Podemos party said Thursday that its 13 members in the lower house have left the governing coalition.
In his statement Thursday, Temer angrily responded to the claim, saying: "I never bought anyone's silence".
Globo reported late on Wednesday that Mr Neves had been recorded asking JBS meat-packing company executive Joesley Batista for 700,000 U.S. dollars (£540,000) to pay for his "Car Wash" defence.
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The investigator wrote that Shelby, "reacted unreasonably by escalating the situation from a confrontation with Mr. Tulsa police have said Crutcher was unarmed and there was no weapon in his vehicle, which was blocking a road.
"If the recordings are confirmed, Temer would have to defend his mandate in multiple fronts: public pressure for his resignation, an impeachment proceeding in Congress and an investigation by the general prosecutor's office", Thomas Favaro, an associate director at consultancy firm Control Risks, told clients in note.
Rousseff was impeached a year ago after the Senate found her guilty of breaking budgetary laws amid the swirling corruption investigation - though she herself was not accused of corruption.
Temer, 76, said he will fight to prove his innocence.
On Tuesday, a congressional panel backed by Brazil's powerful farm lobby called for the dismantling of the Indian affairs agency Funai.
Brazilian Finance Minister Henrique Meirelles called foreign and local investors to try to calm markets, according to a government source briefed on the matter.
"The future will be for soy, corn and sugar cane", said Workers Party lawmaker Nilto Tatto.
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