Earlier, the USA leader - who spent the day in talks with French President Emmanuel Macron - had described Kim as "very open" and "very honorable" - his most positive comments yet about a man he once publicly belittled.
Trump said on Tuesday that Kim "has really been open and I think very honourable based on what we are seeing", noting that the USA and North Korea are already having "good discussions".
"I may hold it up until after a deal is made with North Korea", Trump said in a speech. "Maybe it will be wonderful or maybe it won't".
"In the process of negotiating denuclearization, if North Korea shows concerns about the security of its system, it will be necessary to utilize a declaration of the end of the war", he said.
Trump said he had hoped to meet Kim in May or early June, but a date and venue have yet to be set.
The Punggye-ri nuclear test site, which includes Mount Mantap, is about 400 miles northeast of Pyongyang.
"It would be very easy for me to make a simple deal and claim victory". US officials have said they want to the North to complete full nuclear disarmament.
North Korea condemned US criticism of its human rights record as "ridiculous" as a diplomatic whirlwind accelerated ahead of leader Kim Jong Un's summits with the South and with US President Donald Trump.
Two issues loomed particularly large during the summit, the first of which was North Korea.
The house will act as the summit center. His Senate confirmation hearing for the Australia position had been due to take place Tuesday but was postponed.
There is only a military solution, as much as I truly hate to say it. The aide and other officials requested anonymity to discuss the plans as they were not authorized to discuss them. That could open the door for formal peace talks and a security guarantee from the US - a crucial step to convincing Kim to disarm.
Though there are significant differences between North Korea in 2018 and the former Soviet Union in 1991, the cooperative threat-reduction concept could be a powerful tool to support the verifiable reduction and elimination of North Korea's nuclear arsenal, its other weapons of mass destruction, and their delivery systems.
Harris has spent almost 40 years in uniform and has a reputation as a straight talker. Only a handful of countries have voluntarily given up nuclear programmes and other weapons of mass destruction: South Africa, Iraq, Libya and post-Soviet Ukraine, which signed a deal brokered by Britain and the USA to return nuclear weapons stationed on its soil to Russian Federation.
Cheong Wa Dae spokeswoman Ko Min-jung told reporters, "We tried to ensure that the physical barrier and psychological distance created by 70 years of division will be reduced so that the two leaders could sit face to face and engage in frank dialogue".
Existing trade restrictions have also been extended. The U.S. has announced it will insist that nuclear dismantlement precede economic benefits. There is still uncertainty about what he seeks in return. As the thaw occurred, Mr. Kim conveyed his willingness to talk with Mr. Trump via the South Koreans.
The test caused landslides that physically restructured Mt. Mantap, the location of North Korea's nuclear testing facility.
Kim Jong-un vowed to suspend long-range tests after his dramatic change of policy in recent months.
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