Pompeo, who hails from Kansas, was later asked what they ate. "American beef," he replied.
Kim Yong Chol is a former intelligence chief and vice chairman of North Korea's ruling Workers' Party.
The four-star general has been sanctioned by the Treasury for North Korea's 2014 cyberattack on Sony Pictures, supposedly prompted by the studio's film "The Interview," which ridiculed the country's ruling Kim clan. South Korea claims that Kim Yong Chol was also responsible for the 2010 sinking of a navy ship, which killed 46 sailors.
Kim Yong Chol is also blamed for trading in proscribed conventional arms, and is known as the toughest negotiator at previous nuclear arms talks with the U.S.
Speaking after the dinner, a senior State Department official said the U.S. was still looking for complete and verifiable denuclearization from Pyongyang.
The official, who spoke on the condition of anonymity to brief reporters, said the major goal of the meetings between Pompeo and Kim Yong Chol was to figure out how to persuade North Korea to get rid of its nuclear weapons and missile programs.
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When pressed over how long that could take, the official said: "That's what they're talking about."
U.S. officials and outside experts see these talks as a critical step in determining whether to proceed with the summit between Trump and Kim Jong Un that was originally scheduled for June 12 in Singapore. Trump announced on May 24 that he was calling off the meeting but has since softened his tone.
The potential summit between @POTUS and Chairman Kim presents #DPRK with a great opportunity to achieve security and economic prosperity. The people of #NorthKorea can have a brighter future and the world can be more peaceful.
The State Department official added that Trump would consider a summit of more than one day. Trump is "looking for something historic," he said. But if that is not possible, the U.S. will revert to its maximum pressure campaign on North Korea, the official added.
Asked what had changed to put the talks back on the table since Trump's letter, the State Department official pointed to a follow-up letter from North Korea's vice foreign minister, Kim Kye Gwan. "It was one of the most conciliatory letters I've ever seen," the official said.
He added that he thought the North Koreans believed that they had erred with their harsh rhetoric earlier last week, before Trump's cancellation of the summit.
The New York meetings are taking place as a separate team of American experts has been meeting in North Korea to hammer out the outlines of a joint summit declaration focusing on the timing and components of what each side means by "denuclearization."
NBC News has been told by three U.S. officials that a recent CIA analysis concluded that Kim Jong Un would never give up the nuclear program his country has been developing for seven decades, a finding that conflicts with statements by Trump that Pyongyang intends to denuclearize in the future.
Separately, a team led by the foremost American scientific expert on the North Korean program, Stanford University Professor Siegfried Hecker, a former director of the Los Alamos weapons laboratory in New Mexico, reported this week that full denuclearization of North Korea's massive nuclear weapons and missile complex would take 15 years, and require an unprecedented inspection and verification challenge.
The short time frame to conclude pre–summit negotiations and logistics is further complicated by the changeover at the State Department: Pompeo, who was confirmed only last month to replace Rex Tillerson, has already sidelined Tillerson's top adviser on Asia, Acting Assistant Secretary of State Susan Thornton. Tillerson fought with White House hardliners to get Thornton nominated to that post.
Thornton, who always traveled with Tillerson, was noticeably missing from both of Pompeo's trips to North Korea, as well his New York meeting with Kim Yong Chol.
Pompeo hinted at a shake-up during testimony to the Senate Foreign Relations Committee last week, saying he was "close to making a number of significant announcements about new members of the team, assistant secretary for East Asia affairs and South Asia among them."
There are reports that Thornton's nomination will be withdrawn after the Singapore summit. There is also a major vacancy in Seoul, South Korea: There's no confirmed U.S. ambassador, although Adm. Harry Harris, currently the head of the Pacific Command, has been nominated for that post.
Coordinating the multipronged summit is also challenging because of allied concerns: Japan's Prime Minister Shinzo Abe is seeking reassurance that potential normalization of relations between North and South Korea will not undermine his country's strategic interests. He is scheduled to meet with Trump in Washington next week.
Pompeo and Kim Yong Chol are also due to have additional meetings on Thursday.