Mr. Pompeo denied that the talks went poorly, telling pool reporters that progress was made "in every element of our discussions".
Kim expressed deep appreciation for "the energetic and extraordinary efforts made by Your Excellency Mr. President and for the improvement of relations between the two countries and the faithful implementation of the joint statement".
The message came after Secretary of State Michael Pompeo returned from his third visit to North Korea. "We will be ready", she told reporters aboard Pompeo's plane, as the top USA diplomat flew home to Washington after attending a North Atlantic Treaty Organisation summit.
There was no indication of whether the North Koreans intentionally blew off the meeting.
South Korea's Yonhap news agency, citing unnamed sources, said the North requested talks at a higher level.
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The UNC told the US defense ministry about North Korea's proposal and is waiting for Washington's answer, the source added.
US President Donald Trump met North Korean leader Kim Jong Un on Sentosa Island last month.
"We have to see the USA defense ministry's response, but the UNC reportedly gave a positive signal to North Korea, so there is a good chance of the talks taking place on Sunday", the source said. A total of 7,697 personnel remain unaccounted for from the Korean War.
The US military also announced that 100 wooden coffins had been dispatch to Panmunjom to receive the remains, although repatriation has yet to take place.
Trump himself has remained upbeat about the outcome of the first summit between the leaders of the US and North Korea.
But skepticism remains over whether North Korea's findings will satisfy Japan.
Following the North Koreans' no-show at Thursday's meeting, the U.S. state department said the two delegations had agreed to meet on 15 July.
The source said that Kim did not see any need to hold dialogue with Japan before his summit with South Korean President Moon Jae-in in April, instead favoring relations with the United States, China and Russian Federation.