Kim Yong Chol is the most senior North Korean official to meet top officials for talks in the United States since Jo Myong Rok, a vice marshal, met then-President Bill Clinton at the White House in 2000.
The latest conciliatory declarations capped a turbulent few days of diplomatic brinkmanship that had sent tensions soaring.
U.S. and North Korean officials have continued to meet for pre-summit negotiations in Singapore, while the United States invited Kim Yong Chol to a meeting on American soil, the type of which has not taken place for decades. The North Koreans complained about administration officials talking up the so-called "Libyan model", a scenario that ended with the death of that country's leader.
"It's moving along very nicely", Trump told reporters when asked for an update.
"I think there is a lot of catch up, as in the past these things were usually bottom-up - the groundwork is done ahead of time - but this is all sudden, due to Trump's impulsive nature", said Dan Pinkston, a Seoul-based worldwide relations expert at Troy University.
Mr Pompeo and North Korea's former military intelligence chief Kim Yong Chol met at the three-bedroom apartment residence of the USA deputy ambassador to the United Nation.
"At the same time as working to achieve the goal of denuclearization, we should also build long-term and effective initiatives to keep peace on the Korean peninsula", China's foreign ministry Hua Chunying said in Beijing.
White House Press Secretary Sarah Huckabee Sanders said Wednesday that the Trump administration is continuing preparations to make the summit a reality.
-North Korea summit, the White House said.
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US team heading to Singapore to work on summit
Trump dismissed the so-called Libya model. "We'd like to do it", Trump said. "We'll see what happens - it could even be the 12th". He said meetings were ongoing and that the denuclearization of the Korean peninsula "would be a great thing for North Korea".
Secretary of State Mike Pompeo met with the delegation in NY, and the president said he is hopeful they will travel to Washington DC on Friday.
"Between now and if we're going to have a summit, they're going to have to make clear what they're willing to do", said the official.
"We're going to continue to shoot for the June 12th and expect to do that", she added, referring to the original date scheduled for the summit. The US and China are both signatories to the armistice.
The key to dismantling the sprawling atomic complex, begun six decades ago, Dr Hecker added, "is to establish a different relationship with North Korea where its security rests on something other than nuclear weapons".
Kim Yong-chol and Pompeo will hold two rounds of talks on Thursday between the former spy chiefs on the potential Singapore meeing between Trump and Kim Jong-un. In a letter to Kim last week, Trump abruptly called off the meeting, citing escalating rhetoric from North Korea.
Within hours, however, Pyongyang signaled a continued interest in holding talks, and Trump and his aides hinted that the meeting was back on.
TRT World's Leone Lakhani has more from Washington DC. The envisioned meeting between Japan and North Korea would likely take place between August 1 and 4 when the two ministers are expected to be in Singapore for the ASEAN Regional Forum, the sources said.
Senior officials of the two countries, which have no diplomatic ties, were separately dispatched to the truce village of Panmunjeom, which divides the two Koreas, and Singapore for meetings to narrow down substantive and logistical issues ahead of the planned summit.
He vowed Russia's support for denuclearization and a broader effort to create a stable and long-lasting peace in the region, but indicated that Moscow believes sanctions can be eased while the process is in progress, which diverges from the us position that denuclearization must come first.