US President Donald Trump on Friday warned "madman" North Korean leader Kim Jong-Un that he would be "tested," the latest in an exchange of insults between the two.

"Kim Jong Un of North Korea, who is obviously a madman who doesn't mind starving or killing his people, will be tested like never before!" Trump tweeted.

Trump "insulted me and my country in front of the eyes of the world and made the most ferocious declaration of a war in history", Kim said, according to the official Korean Central News Agency. He blasted Trump's maiden speech to the UN General Assembly as "unprecedented rude nonsense".

Trump branded Kim "Rocket Man" and threatened to "totally destroy North Korea" in his UN speech. Washington followed on Thursday by announcing tougher sanctions on North Korea.

Trump had earlier threatened Pyongyang with "fire and fury."

Trump has traded tough rhetoric with Pyongyang as it pushed an increasingly brazen weapons program in recent months, with missile launches and its sixth and largest nuclear test.

On the fringes of the UN General Assembly in New York, North Korean Foreign Minister Ri Yong-ho told reporters Pyongyang might now consider detonating a hydrogen bomb outside its territory.