The US has dismissed a statement by North Korea accusing Washington of declaring war on the country, calling the idea “absurd”.
The White House also warned Pyongyang to stop provocations after it said it had the right to shoot down US bombers.
A UN spokesman said fiery talk could lead to fatal misunderstandings.
South Korea has called for a level-headed response, warning that accidental clashes in the region could quickly spiral out of control.
North Korean Foreign Minister Ri Yong-ho told reporters on Monday that “the whole world should clearly remember it was the US who first declared war on our country”.
Speaking as he left New York after the UN General Assembly, he said his country had the right to shoot down US warplanes even if they were not in North Korea’s airspace.
His statement came two days after US warplanes flew close to North Korea’s coast in a show of force.
It is not the first time that North Korea has used the phrase “a declaration of war” in relation to the US. Mr Ri’s remarks are the latest in an increasingly angry war of words between Washington and Pyongyang.
Later on Monday, White House spokeswoman Sarah Huckabee Sanders denied the US had declared war, calling the suggestion “absurd”.
Pentagon spokesman Col Robert Manning reacted by saying: “If North Korea does not stop their provocative actions, you know, we will make sure that we provide options to the president to deal with North Korea.”
South Korea called for “astuteness and steadfastness” in responding to what it describes as continued provocations by North Korea. Speaking in New York, the country’s foreign minister, Kang Kyung-wha, called for the prevention of any “further escalation of tensions, or any kind of accidental military clashes in the region which can quickly spiral out of control”.
South Korea’s intelligence service said Pyongyang was readjusting the position of its military aircraft and strengthening its coastal defences, according to the South’s news agency Yonhap.
Stéphane Dujarric, a spokesman for UN Secretary General António Guterres, said that “fiery talk can lead to fatal misunderstandings” and that “the only solution for this is a political solution”.
“We want things to calm down,” China’s ambassador to the UN, Liu Jieyi, told Reuters. “It’s getting too dangerous and it’s in nobody’s interest.”
Despite weeks of tension, experts have played down the risk of direct conflict between the two countries.
North Korea has continued to carry out nuclear and ballistic missile tests in recent weeks, in defiance of successive rounds of UN sanctions.
The country’s leaders say nuclear capabilities are its only deterrent against an outside world seeking to destroy it.
After the North’s latest and most powerful nuclear test earlier this month, the UN Security Council approved new sanctions on the country.
Actions not words
By Jonathan Marcus, BBC News diplomatic correspondent
The rhetoric on both sides may have got out of hand already but the real question is what practical consequences might ensue from the war of words between Washington and Pyongyang?
It should be remembered that the Korean peninsula is not at peace – the Korean conflict of the 1950s was only brought to a halt by an armistice, not a peace treaty.
But it is actions that are likely to provoke renewed fighting, not just words.
The latest North Korean threat to shoot down US warplanes comes in the wake of a recent US patrol that took its B1-B Lancer bombers and their accompanying F-15 fighter escorts over waters to the east of North Korea – the furthest north US warplanes have flown for several months, albeit still outside Pyongyang’s airspace.
The US believes it has every right to do this but if one day Pyongyang judges that these aircraft are on an offensive mission – what then?