"They point the finger at others trying to escape punishment. I warn the criminals that they will not go unpunished," Temer said in a video broadcast on social networks before he left Brasilia for Moscow and Norway.
In the message, Temer did not explicitly mention Joesley Batista, main shareholder in meatpacking company JBS, but his announcement came days after the businessman made an outspoken attack on the president.
As Temer left on a trip aimed at attracting investment, his lawyers filed the lawsuits. One concerns crimes of "slander, defamation and insult" and another is a civil suit, seeking recompense for "moral damages".
Batista caused an uproar last month when he handed to authorities, in connection with a wide-reaching corruption scandal, an audio recording in which Temer appeared to condone payment of hush money to a former lawmaker now in prison.
More From This Section
"It's the country's largest and most dangerous criminal organization. And the president is its leader," Batista said in an in-depth interview with the Epoca weekly published Saturday.
"Those who aren't in prison today are in the Planalto presidential palace. They are very dangerous people. I didn't have the courage to confront them."
Batista agreed to cooperate in exchange for avoiding a conviction.
The tycoon's explosive revelations came after the Supreme Court set in motion corruption and graft probes targeting the president, increasing calls for him to step down.
Among the claims, Batista says tens of millions of dollars were paid to various political parties, including Temer's center-right Brazilian Democratic Movement Party.
Temer, 76, denies the allegations and has insisted he will remain in office.
Following the interview, Temer's office called the allegations a "bunch of lies", saying the president would launch a lawsuit.
"Joesley is one of Brazil's most notorious and successful criminals," it said in a statement.