The US has dismissed a statement by North Korea accusing Washington of declaring war on the country, calling the idea "absurd".
The White House on Monday 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 aircrafts 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.