Mr Cunha is a rival of President Dilma Rousseff, He will be charged with bribery connected to contracts with the state oil company, Petrobras.
The court will rule later on whether Mr Cunha should leave his post. He denies wrongdoing and refuses to stand down.
Mr Cunha has been leading efforts to impeach President Rousseff.
The lower house's Ethics Committee has also approved an investigation into whether Eduardo Cunha should lose his position for lying to a congressional hearing by denying he held bank accounts overseas.
Swiss prosecutors say they later located accounts held by Mr Cunha.
Prosecutors have said Mr Cunha accepted $5m (Â£3.5m) in bribes between 2006 and 2012 in connection with the construction of two Petrobras drilling ships.
He is charged with corruption and money-laundering. The charges are part of a wide-ranging and complex investigation into a huge corruption scheme at Petrobras.
Under Brazilian law, charges against federal congressmen and other top government officials can only be filed and judged by the Supreme Court.
The Brazilian Supreme Court has voted to back corruption charges against the speaker of the lower house of Congress, Eduardo Cunha.