The allegations are the latest twist in a sprawling corruption scandal that has reached the top levels of the country's financial and political elite.
The former presidential candidate denied any wrongdoing stating that the request was on a personal level, as a loan, and that no impropriety was involved.
If Temer resigns or is impeached, Brazil's constitution calls for Congress to name a successor within 30 days. Crowds also gathered outside the presidential palace chanting "Fora Temer" (Temer out).
Brazil's political crisis deepened on Thursday as government allies began openly discussing scenarios for the replacement of embattled President Michel Temer.
Brazilian President Michel Temer says he won't be resigning over allegations that he endorsed hush money payments to a former ally, denying the charges in an address on TV.
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.
Associated Press writer Peter Prengaman reported this story in Rio de Janeiro and AP writer Mauricio Savarese reported from Sao Paulo.
The crisis leaves Temer, 76, facing two immediate problems.
"I can't see how Temer survives this", said David Fleischer, a political science professor at the University of Brasilia. More than 80 people have been charged with bribery and money laundering, according to figures cited by CNN.More news: Benzema slams Valbuena over sex-tape saga
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 NY trading as investors ditched Brazilian assets at the sight of more political turmoil. The real also fell 7% against the USA dollar.
Images broadcast by the Globo News television station on Thursday showed police entering a beachfront apartment, as well as searching offices in congress belonging to Neves and two legislators from the ruling PMDB party. Aecio Neves, who almost won the presidency in 2014 and planned to run again next year.
Leading lawmakers and a third of Temer's cabinet have already been caught up in an investigation of systematic bribery in return for political favors and contracts with state-run enterprises.
Brazilian President Michel Temer late Wednesday denied a report that he endorsed bribing a now jailed former congressman to keep him quiet, an allegation that could have widespread impact in a nation swamped by graft scandals and a populace furious about it.
He spearheaded the impeachment of former president Dilma Rousseff past year.
The tape reportedly records the President saying, "You need to keep this up, got it?"
In an official statement, Temer admitted to meeting with Batista, but said he never tried to buy Cunha's silence. JBS declined to comment immediately.
In a March interview with Reuters, Judge Mendes - who is strongly linked to a party that is allied with Temer's - said that the president would not necessarily lose power if the election were annulled because he was not the head of the ticket.