In a tweet, Trump called Kim "a madman who doesn't mind starving or killing his people."
Trump's latest tirade came after the North Korean dictator chastised Trump, calling the president's United Nations address earlier this week "unprecedented rude nonsense."
"The mentally deranged behavior of the U.S. president openly expressing on the UN arena the unethical will to 'totally destroy' a sovereign state, beyond the boundary of threats of regime change or overturn of social system, makes even those with normal thinking faculty think about discretion and composure," Kim said in a statement circulated by state news agency KCNA.
Early Friday, Kim said that Trump's remarks at the U.N. merit the "highest level of hard-line countermeasure in history."
The heated rhetoric comes as North Korean Minister of Foreign Affairs Ri Yong Ho said that his country may consider a test of a hydrogen bomb in the Pacific Ocean, according to a report from South Korea's official news agency Yonhap.
Ri said the potential test of "the most powerful detonation of an H-bomb" would be one possible "highest-level" action against the U.S., according to the report.
— CNBC's Christine Wang and Leslie Shaffer contributed to this report.