Earlier, Kim directed his own insults at Trump, whom he called a “mentally deranged U.S. dotard."
The back-and-forth between the United States and North Korea is the latest in a series of events escalating tensions between the two nations.
Trump ratcheted up the rhetoric against North Korea during a U.N. General Assembly meeting in New York this week, calling on allies, including China, to do more to isolate Kim.
The president threatened that the United States would be forced to "totally destroy" North Korea, if the United States or its allies are "forced to defend" themselves against the regime.
Asked Friday morning about whether the president's latest tweets were working, Secretary of State Rex Tillerson highlighted the U.S. diplomatic efforts to address North Korea.
"The president obviously takes the responsibility of the security of the American people very seriously and it is his first and foremost responsibility," Tillerson said on ABC News's "Good Morning America." "But we're not in this alone. We have developed strong allies and strong alliances internationally, and we are engaging with North Korea's most important supporters -- economic supporter their friends China and Russia, to have them also engage with Kim Jong Un."
"We will continue our efforts in the diplomatic arena, but all of our military options -- as the president has said -- are on the table," he added.