Trump made the remarks on Twitter Friday morning.
"Kim Jong Un of North Korea, who is obviously a madman who doesn't mind starving or killing his people, will be tested like never before!" the president tweeted.
Trump's remarks follow new rhetoric from Kim Thursday, in which the leader called the U.S. president a "dotard" and lambasted his fiery address to the U.N. General Assembly on Tuesday. In that speech, Trump said North Korea would be destroyed if it took action against the United States.
"The mentally deranged behavior of the U.S. president openly expressing on the U.N. arena the unethical will to 'totally destroy' a sovereign state ... makes even those with normal thinking faculty lose discretion and composure," Kim said.
Japan's government said Friday it is also concerned with remarks by North Korea's foreign minister that the regime could test a hydrogen bomb in the Pacific.
"We cannot deny the possibility it may fly over our country," Japanese Defense Minister Itsunori Onodera said.
North Korea conducted its sixth and largest nuclear test on Sept. 3, which it said was an H-bomb.