The White House dismissed the claim as “absurd”.
The latest threats stoked a week-long war of words that began when the American leader threatened, in his first address to the United Nations General Assembly, to “totally destroy” North Korea if it launches an attack.
Alarm over Pyongyang’s nuclear and ballistic missile programs dominated this year’s gathering of world leaders at the UN, amid fears the heated rhetoric could accidentally trigger a war.
Those fears were further heightened after US bombers flew off the coast of North Korea on Saturday.
Speaking to reporters outside his New York hotel, Foreign Minister Ri Yong-ho responded to Trump, who had warned on Twitter at the weekend that North Korea “won’t be around much longer” if it keeps up its threats.
Ri said the international community had hoped that the “war of words” between the two countries would “not turn into real actions.”
“However, last weekend, Trump claimed our leadership would not be around much longer,” Ri said. “He declared a war on our country.”
“Since the United States declared war on our country, we will have every right to take counter-measures including the right to shoot down US strategic bombers even when they are not yet inside the airspace border of our country,” he said.
“The question of who won’t be around much longer will be answered then.”
Ri noted that the statement from Trump was “clearly a declaration of war” even if it came from the US president. Under the US Constitution, Congress is the sole power that can declare war.
The White House disputed the foreign minister’s interpretation of Trump’s sabre rattling.
“We have not declared war against North Korea and frankly the suggestion of that is absurd,” said White House spokeswoman Sarah Huckabee Sanders.
In his UN address last week, Trump threatened to “totally destroy” North Korea and derided leader Kim Jong-un as a “rocket man” who was “on a suicide mission.”
Kim hit back with a personal attack on Trump, branding him “mentally deranged” and a “dotard” and warning he would “pay dearly” for his threat.
Russia’s Foreign Minister Sergei Lavrov has compared the past week’s rhetoric to a “kindergarten fight between children” and had urged the “hot heads” to calm down.
Asked about the North Korean minister’s latest remarks, UN spokesman Stephane Dujarric acknowledged the dangerous spike in tensions, saying: “When you have the rise of tension, the rise of rhetoric, so does the risk of miscalculation.”
North Korea in recent weeks detonated its sixth nuclear bomb and has test-fired intercontinental missiles — saying it needs to defend itself against hostility from the United States and its allies.
During his address to the General Assembly on Saturday, Ri launched a personal attack on Trump, calling him a “mentally deranged person full of megalomania.”
And just hours before Ri took the UN podium, US bombers flew off the east coast of North Korea, the furthest north of the demilitarized zone that any US aircraft has flown this century.
The Pentagon described the mission as a “demonstration of US resolve and a clear message that the president has many military options to defeat any threat.”
A Pentagon spokesman stressed on Monday that the bombers flew in international airspace. “So we have the right to fly, sail and operate and we are legal permissible around the globe,” he said.
On Saturday, Ri warned that Trump’s threat to destroy North Korea made “our rockets’ visit to the entire US mainland all the more inevitable.
The bellicose rhetoric comes as international alarm mounts over Pyongyang’s weapons ambitions — including a suggestion by Ri last week that the country is considering detonating an H-bomb over the Pacific.
US Defense Secretary Jim Mattis has said such a move would be a “shocking display of irresponsibility.”
As tensions over North Korea reached fever-pitch, Japanese Prime Minister Shinzo Abe called a snap election.
The North Korea crisis appears to have given the hawkish Abe a welcome boost in the polls following a series of scandals.