The spokesman apparently rejected the United States demand and reiterated Pyongyang's call for a "phased" and "synchronous" approach, saying that it would be the shortest way to realize a nuclear-free Korean Peninsula, Yonhap news agency reported.
Pompeo said the two sides agreed to hold discussions on July 12 on repatriation, and also discussed "modalities" for the destruction of a missile engine testing facility.
"I would like to pay my tribute to the strong leadership you have demonstrated in negotiating with North Korea", Abe told Pompeo.
The danger now is that if happens if Trump finds that North Korea isn't willing to bend to his most unrealistic demands, the countries could find themselves in an even more unstable position than where they started.
However, in another twist of approach towards Kim - who Trump once called "Little Rocket Man" - the United States administration renewed its sanctions on North Korea for another year.
Pompeo has the crucial task of dispelling growing skepticism over how seriously Kim is about giving up his nuclear arsenal and translate the upbeat rhetoric following the first meeting between leaders of the US and North Korea into concrete action. The US has also stopped using the catchphrase of "complete, verifiable, irreversible denuclearisation" of North Korea that it had insisted upon happening before North Korea gets any relief from a crippling sanctions regime.
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Now, three weeks later, the two sides were still at odds on all issues, including exactly what denuclearization means and how it might be verified, after a third visit to Pyongyang by U.S. Secretary of State Mike Pompeo.
However, he emphasized that the North still has trust in the US President, apparently hinting that Pyongyang wanted to keep the ongoing denuclearization talks on track.
Pompeo, however, said that more talks were needed on both.
"I think we made progress in every element of our discussions", he added.
Kim later said that "there are things that I have to clarify" to which Pompeo responded that "there are things that I have to clarify as well".
A failure in the talks would amount to an embarrassment for Pompeo, whom Trump tasked with leading the negotiations and who had said repeatedly that Kim Jong Un was ready to commit to something no North Korean government had delivered on in decades.
He characterized the discussions as positive, but hours after he left Pyongyang, North Korea issued an angry statement rejecting what it described as Washington's "gangster-like" demand for rapid nuclear disarmament.
Pompeo's visit to North Korea has led to the launch of working groups to deal with details such as verification. "Some places a great deal of progress, other places there's still more work to be done", he said, according to a pool report from us reporters who accompanied him to Pyongyang. In practical terms, however, he only mentioned that officials from both sides would meet on July 12 as a working group to discuss the repatriation of the remains of some USA soldiers killed during the 1950-1953 Korean War. South Korean news outlet, The Chosun Ilbo, referred to an unconfirmed "source in Washington" which stated that the "Rocket Man" CD came up in conversation during the historic summit, after Trump learned Kim had never heard the hit song.