The accusations by Attorney General Rodrigo Janot relate to the release this week of an audio recording that allegedly shows the president tried to derail a massive corruption investigation known as 'Car Wash.'
In the secretly recorded conversation, first reported by the O Globo newspaper on Wednesday evening, the president is purported to have given his approval to monthly payments of hush money to a jailed politician.
That politician, Eduardo Cunha, was once a powerful member of Temer's ruling party and is now in prison for taking bribes in the anti-corruption drive.
Cunha has previously said that he has compromising information about other politicians linked to the bribery scandal.
The scandal threatens to derail Temer's presidency just a year after the center-right politician took over from impeached leftist President Dilma Rousseff.
Temer denied any wrongdoing in a televised address Thursday and rejected calls for him to resign.
Thousands of people demonstrated against Temer on Thursday in the capital, Brasillia, and in Rio de Janeiro. More protests are planned for the weekend.
Temer faces an investigation by the Supreme Court as well as the potential collapse of his base in Congress, opening the way to impeachment proceedings.
So far, Temer's PMDB party and coalition partner, the PSDB Social Democrats, are still backing the president, with the only calls for impeachment coming from opposition lawmakers.