"We'll be having a second summit with Chairman Kim in the not too distant future", the president added.
Prior to a second Trump-Kim summit the Trump Administration "should demand tangible progress by North Korea toward denuclearization, such as a declaration of WMDs and facilities, permitting onsite inspections, a clear commitment and timetable to end production and destroy weapons of mass destruction or other steps", says Heritage Foundation Senior Research Fellow Brett Schaefer.
"Kim Jong Un wrote a letter, a lovely letter, asking for a second meeting, and we will be doing that".
Trump says Secretary of State Mike Pompeo (pahm-PAY'-oh) is working out the details for the second Trump-Kim meeting.
Denuclearization talks between the United States and North Korea had stalled since Trump called off Pompeo's scheduled visit to Pyongyang in July, citing what he called a lack of progress in the North Korean denuclearization process.
Mr Trump referred to Mr Kim's "tremendous enthusiasm". Trump has a robust schedule during his stay in NY, including meetings with Israeli Prime Minister Benjamin Netanyahu and British Prime Minister Theresa May.
Moon, speaking through a translator, said the two countries had made "modifications" to the deal.
"When the president meets with North Koreans", she said, "that's in itself a bonus.so until we find out a little bit more, the president won't be meeting with them".More news: Juventus midfielder Matuidi: Ronaldo dismissal was incomprehensible
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To prepare for that trip, Pompeo offered to meet with North Korean Foreign Minister Ri Yong Ho this week at the U.N. General Assembly in NY, but the North Koreans have still not accepted that meeting.
"We'll be doing that", he said.
USA intelligence agencies have said that North Korea, which has not tested a ballistic missile since agreeing to the summit months ago, continues to work on technical advancements in its nuclear arsenal in secret.
Still, Pompeo said Monday he will head to Pyongyang soon for those conversations, which need to happen to see if the two sides are to be on the same page, U.S. Ambassador to the United Nations Nikki Haley told ABC News on Sunday.
If there is a through line to the still-evolving Trump doctrine on foreign policy, it is that the president will not subordinate American interests on the world stage, whether for economic, military or political gain.
Trump's address to the General Assembly comes Tuesday, and on Wednesday he will for the first time chair the Security Council, with the stated topic of non-proliferation of nuclear, chemical and biological weapons.
"That was a very risky time". On Monday he is also set to participate in a Global Call to Action on the World Drug Problem and then, in addition to Moon, meet with Egyptian President Abdel-Fattah el-Sissi and French President Emmanuel Macron.
This year's General Assembly will be Mr. Trump's second meeting since taking office and Pompeo's first since taking over as secretary of state. "This is one year later, a much different time".
Mr. Moon is expected to convey to Mr. Trump a personal message from Mr. Kim delivered at their inter-Korean talks last week.