The US has said a statement from North Korea accusing Washington of declaring war on it was “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”.
His comments were a response to a tweet from President Donald Trumpsuggesting North Korea would not “be around much longer” if its leaders continued their rhetoric.
Two days ago US warplanes flew close to North Korea’s coast in a show of force.
Speaking as he left New York after the UN General Assembly, Mr Ri said his country had the right to shoot down US warplanes even if they were not in North Korea’s airspace.