How has Mr Pompeo responded?
Abraham Denmark, a senior defense official for East Asia under former President Barack Obama, said: "This is a rejection of USA demands for unilateral denuclearisation by North Korea, and a clear message that the US will need to give up more to make progress".
Kim agreed that the work was important.
Kim did not comment on the issue of denuclearisation, saying only that there "are things I have to clarify".
Speaking later to members of the US-Vietnamese business community in Hanoi, Mr Pompeo called on Mr Kim to replicate Vietnam's "miracle" of economic growth by improving ties with the US, vowing that America keeps its promises with former foes. "Some places a great deal of progress, other places there's still more work to be done", he said.
Those include the formation of working group to determine exactly how North Korea's denuclearisation will be verified and a Thursday meeting with Pentagon officials to discuss the return of remains of Americans soldiers killed during the Korean War. Instead, Pompeo sat down with Kim Yong Chol, the country's spy chief, for three hours. Nauert told reporters that the two sides spoke during a dinner following their almost three-hour meeting Friday night and they were "cracking jokes" and "exchanging pleasantries".
Mr Pompeo arrived in Pyongyang on Friday for a two-day visit - his third trip to the isolated state since taking the secretary of state job in April. We can hurt them more than they'll hurt us.
"We expected that the U.S. side would come with productive measures conducive to building trust in line with the spirit of the North-U.S. summit and [we] considered providing something that would correspond to them", an unnamed foreign ministry spokesman said, according to the South Korean news agency Yonhap.
Pompeo wanted to agree on at least an initial list of nuclear sites and an inventory that could be checked against the available intelligence, US intelligence officials told Reuters earlier.
Both issues are considered essential tests of whether Kim is serious about talks.
Some analysts and lawmakers have expressed alarm that the talks appear to have run into difficulties, although others see a possible North Korean negotiating ploy.
Today's meeting follows three hours of talks at the Baekhwawon state guest house in Pyongyang on Friday, following the USA delegation's arrival in Pyongyang just before noon. It wasn't my language, it was the language of Chairman Kim, he committed to complete denuclearisation.
Some U.S. experts said the disputes highlight the risk of Trump granting premature concessions to North Korea.
In criticising the talks with Pompeo, however, North Korea carefully avoided attacking Trump, saying "we wholly maintain our trust toward President Trump", but also that Washington must not allow "headwinds" against the "wills of the leaders". The terse statement, a US source said, was expected.