The Supreme Court gave its green light to the investigation, the state-owned Agencia Brasil and leading newspapers reported.
The veteran centre-right politician, who took over last year with a promise to restore Brazil's stability after the impeachment of leftist president Dilma Rousseff, remained defiant.
He was expected to speak to the nation later today and "he will explain himself", a spokesman told AFP. "He is sure that he acted in a correct manner."
Temer is deeply unpopular but has been able to rely on a dominant alliance between his centre-right PMDB party and the PSDB Social Democrats, along with a coalition of smaller parties.
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Less than 24 hours after O Globo newspaper's explosive report that Temer had been caught on tape agreeing to bribe the jailed politician, he faced three formal requests for his impeachment.
More worryingly for Temer, the PSDB appeared to be losing faith. "If the evidence is confirmed then we will ask our (ministers) to leave the government," lower house deputy Ricardo Tripoli, a party leader, told AFP.
"Brazil has no government," Rodrigues said.
The Sao Paulo stock market's Bovespa index crashed more than 10 per cent after opening, triggering an automatic suspension of trading for 30 minutes.
Temer was reported late yesterday by O Globo to have been secretly recorded agreeing to payments of hush money to Eduardo Cunha, the disgraced former speaker of the lower house of Congress.
According to the report, the president discussed the matter with Joesley Batista, an executive from the meatpacking giant JBS, on March 7.
Batista told Temer that he was paying money to make sure that Cunha -- thought to have encyclopedic knowledge of Brazil's notoriously dirty political world -- would keep quiet while serving his jail sentence for taking bribes.
Temer's office issued a statement saying: "President Michel Temer never solicited payments to obtain the silence of former deputy Eduardo Cunha."
Globo did not say how it got the information about the recording, which it said was offered in a plea bargain between Batista and his brother Wesley with prosecutors.
A separate secret recording made by Batista allegedly caught Senator Aecio Neves, head of the PSDB party and a close Temer ally, asking him for a bribe of two million reais, or around USD 600,000.
The Supreme Court suspended Neves. Officers could be seen entering Neves' property in Rio de Janeiro and his sister Andrea was arrested in Belo Horizonte.
The scandal is the latest shockwave from the "Car Wash" graft probe ripping through Brazilian politics.
Investigators have uncovered a massive scheme in which politicians took bribes in exchange for getting big businesses over-inflated contracts with state oil company Petrobras. The bribery and embezzlement then rippled far beyond, pulling in many of the country's most famous executives and leaders.
Former president Luiz Inacio Lula da Silva, a leftist icon for many in Latin America, faces five corruption trials, while a third of the Senate and a third of Temer's own cabinet are under investigation.
Although alleged to have participated in large-scale bribery deals, he could not be prosecuted for crimes prior to his mandate. However, the hush money claims date from well within his time in office.
Even before the latest crisis, Temer was mired in controversy.
Rousseff and her leftist allies accuse him of having engineered her impeachment and his own rise to power last year in what they say amounted to a coup d'etat.
Rousseff, from the leftist Workers' Party, was found guilty by Congress of having illegally manipulated government accounts to hide the true extent of Brazil's financial woes.