Pompeo, who held talks with Kim over the weekend, said "we have a pretty good understanding between our two countries about what the shared objectives are".
But both Kang and Pompeo insisted that they agreed on the need for the "total, permanent and verifiable" denuclearization of the divided peninsula.
Kim had revealed plans to shut down the country's northeastern testing ground by the end of May during his summit with South Korean President Moon Jae-in last month.
The Americans had boarded Pompeo's plane out of North Korea without assistance and then transferred in Japan to the Boeing C-40 outfitted with medical facilities for the trip back to the U.S.
The United States and North Korea are technically still at war - a stop-gap armistice ended the brutal three-year Korean war in 1953 and around 30,000 American troops remain in neighboring South Korea, which the U.S. supported in the conflict.
North Korea's state-run media have accused the Japanese government of attempting to hinder the peace process by bringing up the abduction issue that has already been settled.
State Department officials refused to discuss anything but the most basic details of their conditions, citing privacy concerns in keeping with the minimal amount of information they had released since the men were imprisoned.
Over the past year, Kim and Trump have added a personal touch to a half-century of worldwide enmity, swapping insults and both openly threatening direct military action.
Pompeo traveled to North Korea for the second time this week and returned early on Thursday morning with three American citizens freed from prison in North Korea. It will be the first meeting ever between a sitting US president and the leader of North Korea.
Iran targets Israeli bases across Syrian frontier, Israel pounds Syria
Iran and Israel have engaged in a number of clashes in Syria, where Tehran's forces are aiding Bashar al-Assad's regime. German Chancellor Angela Merkel and French President Emmanuel Macron called for a de-escalation between the two sides.
The Trump-Kim summit is an achievement in and of itself. "It will make it more hard for Kim Jong Un to deny inspections now that he has placed them on the table".
North Korea said Saturday that all of the tunnels at the country's northeastern nuclear test site will be destroyed by explosion in less than two weeks, ahead of Kim's summit with Trump. But he said he thinks Kim has a good understanding of what the USA means by all of that.
"Make no mistake about it, America's interest here is preventing the risk that North Korea will launch a nuclear weapon into L.A. or Denver or to the very place we're sitting here this morning", Pompeo said from Washington.
Watch Pompeo's opening statements at the press briefing below..
We are closing out a dramatic week for diplomacy in Asia. "So I hope this time is different", said Gardner.
And Pompeo said the United States would remain on board to help develop the North's economy, which has been devastated by its own mismanagement and crippling worldwide sanctions. Of the three, Kim Dong Chul had been held the longest.
However, his comments made clear that the two sides remained far apart on the key issue of what they mean by denuclearisation.
Hardliners in the North are believed to see a nuclear weapon as a guarantee against outside efforts to topple the regime, but Trump insists he will not tolerate their growing arsenal.