Meanwhile, the United Nations chief has warned that the fiery talk could lead to misunderstandings.
Speaking to reporters in New York, North Korean Foreign Minister Ri Yong Ho claimed: "The whole world should clearly remember it was the U.S. who first declared war on our country". Trump also used the derisive reference to Kim in his speech to the U.N. General Assembly on September 19.
The latest escalation in the war of words between the nations came two days after US Air Force bombers, escorted by fighter jets, flew over waters east of North Korea on Saturday.
Trump over the weekend dismissed Ri's speech at the United Nations, declaring: "If he echoes thoughts of Little Rocket Man, they won't be around much longer!" "Even the fact that this comes from someone who is now holding the seat of the USA presidency is clearly a declaration of war".
Ri Yong Ho told reporters in New York: "The whole world should clearly remember it was the United States who first declared war on our country".
"If we take that option it will be devastating - devastating - for North Korea", Trump said in the Rose Garden during a news conference.
Kim responded with the first-ever direct statement from a North Korean leader against a US president, lobbing a string of insults at Trump. -North Korea nuclear crisis, no one can be sure where this would lead or whether the North will even carry out its threat. In a show of might, US bombers and fighter escorts flew Saturday to the farthest point north of the border between North and South Korea by any such American aircraft this century.
Of that flight, Pentagon spokesperson Dana White said, "This is the farthest north of the Demilitarized Zone any USA fighter or bomber aircraft have flown off North Korea's coast in the 21st century".
North Korea, which has remained technically at war with the USA since the 1950-53 Korean War ended in a truce and not a peace treaty, has been working to develop nuclear-tipped missiles capable of hitting the United States mainland and conducted its sixth and largest nuclear test this month.
69 percent of Democrats say Trump is a bigger danger to the US, while 47 percent of independent voters call the president a greater threat and 41 percent cite Kim as more risky.
"This is something I feared", he said.
"It's never appropriate for a country to shoot down another country's aircraft when it's over global waters".
"It's very likely that North Korea will conduct further provocations", Kang said. In 1969, a North Korean fighter jet shot down an unarmed US reconnaissance plane and killed all 31 crewmembers on board.
The Trump administration's stance on North Korea has been hardening in recent months as the North has been stepping up the aggressiveness of its nuclear and missile tests.