Ri Yong-ho said this could apply even if the warplanes were not in North Korea's airspace.
The White House dismissed the statement as "absurd". The Pentagon warned Pyongyang to stop provocations.
A UN spokesman said fiery talk could lead to fatal misunderstandings.
Mr Ri's comments were a response to Mr Trump's tweet that the North Korean leadership would not "be around much longer" if they continued their rhetoric.
"The whole world should clearly remember it was the US who first declared war on our country," Mr Ri told reporters as he was leaving New York, where he had addressed the UN General Assembly on Saturday.
"Since the United States declared war on our country, we will have every right to make counter-measures, including the right to shoot down United States strategic bombers even when they are not inside the airspace border of our country."
Mr Trump's tweet followed Mr Ri's fiery speech to the UN on Saturday, when he described the US president as a "mentally deranged person full of megalomania" on a "suicide mission".