"What is unclear to Chairman Kim Jong Un is not the will for denuclearization, but the concern that if [North Korea] denuclearizes, whether the US can end hostile relations and guarantee the security of the [Kim] regime", Moon said.
"A U.S. delegation is in ongoing talks with North Korean officials at Panmunjom", State Department spokeswoman Heather Nauert said.
"Chairman Kim agreed on that", he added.
Another US team went to Singapore, where the meeting is planned to take place, over the weekend to focus on logistics.
South Korean President Moon Jae-in said on Sunday that practical talks will be held between the us and North Korea "very soon", and the outcome of those talks will dictate if the June 12 summit will go ahead.
A senior administration official said Trump himself dictated the letter, which also warned North Korea that US nuclear power was vastly superior to Pyongyang's.
At least that's what the senior East Asia director on the White House National Security Council, Matt Pottinger, told reporters during a press briefing last week.
"It has a chance to be a great, great meeting for North Korea and a great meeting for the world", he said.
Moon also said that Kim reaffirmed his commitment to working towards a formal end to the 1950-53 Korean War.
Saturday's Korean summit came hours after South Korea expressed relief over revived talks for a Trump-Kim meeting. "And there's no more concrete sign that this is going forward than having that group of, that team of three, as reported by the Washington Post, sitting down with the North Koreans to hammer out those details", said Lee.
Pyongyang has rejected unilateral disarmament and has always couched its language in terms of denuclearisation of the Korean Peninsula.
It is likely that North Korea has proposed shipping in advance certain types of missiles such as intercontinental ballistic missiles capable of hitting the United States.
Despite repeated references to "denuclearization of the Korean Peninsula" by the North, it remains unclear whether Kim will ever agree to fully abandon his nuclear arsenal. It said the Americans met with North Korean Vice Foreign Minister Choe Son Hui.
U.S. officials also entered North Korea yesterday to hold talks on preparations for a possible summit, a U.S. newspaper reported.
A statement from North Korea's state news agency KCNA said Kim expressed "his fixed will" on the possibility of meeting Trump as previously planned.
On Saturday, Trump said that talks in preparation for the June 12 summit in Singapore are now progressing very well.
Seeking a denuclearisation deal with North Korea was a top priority for a leader who came to the White House on a political promise of putting "America first" while withdrawing from global agreements on climate change and Iran. "He's going to run this the way he sees fit".
According to reporting from NPR's Elise Hu, the Saturday meeting between the leaders of the two Koreas happened on Kim's request.