Mr Trump's actions follow his speech at the United Nations general assembly this week, when he escalated his rhetoric against North Korea amid a months-long crisis over Kim Jong Un's expanded missile testing programme.
"I will make the man holding the prerogative of the supreme command in the USA pay dearly for his speech calling for totally destroying [North Korea]", Kim added.
Pyongyang's stated aim is to be able to target the U.S. mainland and the nation has flaunted the advances in its weapons programme in recent weeks, with the September test of what it said was a miniaturised H-bomb capable of being loaded onto a rocket.
In a country where the leader is essentially portrayed as a god, Kim's decision to respond personally to Trump's speech to the U.N. General Assembly and pledge reprisals escalated the standoff over the North's nuclear program in a way that neither he nor his predecessors had done before.
"We have no idea about what actions could be taken as it will be ordered by leader Kim Jong Un", Ri who is scheduled to address the U.N. General Assembly on Saturday said. "Once we can assess the nature of this threat, the President will make a decision regarding the appropriate actions".
"And I think the initiative stand with the United States and South Korea to lessen the tensions by opening up a channel for a negotiation process". "But I continue to call on all those responsible nations to enforce and implement UN sanctions and impose their own measures, like the ones I am announcing today".
Earlier this month, North Korea detonated a powerful hydrogen bomb at its Punggye-ri nuclear test site in the north-east of the country.
He added that he is "thinking hard" about his response and that Trump "will face results beyond his expectation".
The US led a push at the United Nations for tough sanctions that were adopted on September 11, and has imposed unilateral measures to punish firms that do business with North Korea.
Treasury Secretary Steve Mnuchin said he spoke at length with the head of China's central bank but "I am not going to comment on confidential discussions".
Pyongyang conducted its sixth and largest nuclear test on September 3 and has launched dozens of missiles this year as it accelerates a programme aimed at enabling it to target the United States with a nuclear-tipped missile.