"He should have been handled a long time ago by Clinton. I won't mention the Republicans, by Obama," he added.
Trump said that the situation of North Korea should have been handled by now, and added that he will do it now.
"Little rocket man. We're going to do it. Because we really have no choice, we really have no choice," he said.
Earlier in the day, Trump had called Kim Jong Un "a madman" who doesn't mind starving or killing his people'.
"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 said in a tweet.
Trump's tweet comes after Kim Jong Un blasted the US President as "mentally deranged" and vowed to make him pay dearly for threatening to destroy his country.
Trump had recently ordered fresh sanctions over Pyongyang's weapons programmes and even praised China for taking an action to limit financial transactions with North Korea.
As per the new penalties', the nations will have to choose whether to do business with the United States or North Korea.
"North Korea's nuclear program is a grave threat to peace and security in our world, and it is unacceptable that others financially support this criminal, rogue regime," Trump said in a press briefing with the leaders of South Korea and Japan.
He added that United States continues to seek a "complete de-nuclearisation of North Korea."
Trump, in his first address to the UNGA on September 19, attacked North Korea and termed its supreme leader Kim Jong-un as "a rocket man, who is on a suicide mission".
He further said, "It is time for North Korea to realise that de-nuclearisation is its only acceptable future."
Trump asserted that North Korea's reckless pursuit of nuclear weapons threatens entire world with immense loss of human life.
North Korea's Foreign Minister Ri Yong-ho on Wednesday termed Trump's speech at UNGA as "sound of a dog barking."
"There is a saying that goes: Even when dogs bark, the parade goes on. If (Trump) intended to scare us with the sound of a dog barking then he is clearly dreaming," said Ri at the United Nations Headquarters in New York.
The United Nations Security Council had also condemned the recent launch of another ballistic missile by the Democratic People's Republic of Korea (DPRK) and called on the country's leadership to halt such activities and explore resumption of sincere dialogue on de-nuclearisation.
UN Secretary-General Antonio Guterres called on the DPRK leadership "to cease further testing, comply with the relevant Security Council resolutions, and allow space to explore the resumption of sincere dialogue on de-nuclearisation."
Raising concern over the North Korean missile-tests, Chinese Foreign Minister Wang Yi had said that negotiations is the only way to find out the solution and urged the ruling Democratic People's Republic of the Korean Government to not go further in 'dangerous direction'.
Wang said that the situation of North Korea is of utmost importance and is of common concern among the United Nations member countries.
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