Moon and Kim held a surprise summit in the Demilitarised Zone separating their two nations on Saturday in a scramble to save a slated 12 June summit between North Korea and the United States in the city-state of Singapore. North Korean diplomats and officials are not allowed to travel more than a few miles outside New York City. "It will happen!" Trump added.
The White House emphasized that it has remained in close contact with South Korean and Japanese officials as preparations for the talks continue.
US officials cast the on-again, off-again drama as in keeping with Trump's deal-making style, and reflective of the technically still-warring leaders testing each other.
Kim, known as the young leader's right hand man, will become the highest North Korean official to visit the U.S.in almost two decades.
Kim Yong Chol, who previously headed the country's spy agency, has played a prominent role in negotiations with South Korea and the United States in recent months.
Sung Kim, U.S. ambassador to the Philippines, is leading the U.S. delegation hold talks with North Korean officials at the border village of Panmunjom, Sanders said. Pentagon official Randall Schriver is also in Seoul now, the Post said.
"We have put a great team together for our talks with North Korea", Mr Trump said in a Twitter post on Tuesday. North Korea and the United States are still technically at war and have no diplomatic ties because the 1950-53 Korean War ended with an armistice, not a peace treaty.
Moon said Sunday that the North's disarmament could be still be a hard process even if Pyongyang, Washington and Seoul don't differ over what "complete denuclearization" of the peninsula means.
Washington wants to see quick and "irreversible" denuclearization.
Abe stressed that the worldwide community should be united to solve North Korea's nuclear, missile and abduction issues. Pyongyang has rejected unilateral disarmament and has always couched its language in terms of denuclearization of the Korean Peninsula. He said Japan will make efforts for worldwide solidarity.
Pyongyang has claimed its ICBMs are capable of reaching the United States.
Mr Kim's de facto chief of staff, Kim Chang-son, meanwhile, flew to Singapore, the scheduled site of the meeting, via Beijing late on Monday, Japanese public broadcaster NHK reported.
American officials are skeptical that Kim will ever fully abandon his nuclear arsenal.
Mr Kim, in a telling line from a dispatch issued by the North's state-run news service earlier on Sunday, "expressed his fixed will on the historic (North Korea)-US summit talks".
"However, during the U.S".
Trump thinks "ongoing discussions are going very well", White House spokeswoman Sarah Sanders said. The U.S. has also accused Kim of orchestrating the 2014 hack that exposed thousands of emails from executives at Sony Pictures Entertainment.
The officials may discuss bridging the gap between Washington and Pyongyang on what a deal on the North's nuclear weapons would look like.
Officials won't say that the June 12 Singapore summit is back on, but preparations on both sides of the Pacific are proceeding as if it is. A US delegation traveled to the country over the weekend to discuss the possible meeting.
Libyan leader Muammar Gaddafi was deposed and killed by NATO-backed militants in 2011 after halting his nascent nuclear program.