Shortly after the apparent setback was reported, President Trump tweeted the translation of a letter he had received from Mr Kim, in which the North Korean leader said: "I deeply appreciate the energetic and extraordinary efforts made by Your Excellency Mr President".
In the letter, Kim also expressed his appreciation for Trump's efforts toward improving U.S. -DPRK relations and towards the "faithful implementation of the joint statement".
Moon said he believed Trump and Kim would eventually make good on the promise made before the worldwide community.
Doubts over the prospect of a breakthrough were further exacerbated following the secretary of state Mike Pompeo's overnight visit to Pyongyang last week, during which a promised meeting with Kim failed to materialize.
What's next for Secretary of State Pompeo's negotiations with North Korea?
Pompeo also said sanctions would remain in place until "a final fully verified denuclearization" that he said Kim had agreed to occurred.
But no one showed up from the North, South Korea's Yonhap news agency said, citing a government official.
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Defence officials have already shipped dozens of cases to Panmunjom in anticipation of receiving the remains of United States troops.
Following his historic meeting with Mr Kim in Singapore, Mr Trump offered unspecified "security guarantees" to North Korea and said the United States would cease war-game exercises with the South. The North Korean foreign ministry described the talks as "cancerous" and accused Pompeo of using a "gangster-like" approach while negotiating terms to denuclearise. He added that "great progress being made!".
According to the U.S. information, a Russian-flagged tanker identified as Patriot transferred fuel to a tanker on April 10 which then arrived at North Korea's Nampo port five days later.
"Midday today they contacted us and offered to meet on Sunday", Ms Nauert said in a statement.
The meeting was to focus on the return of remains of America's war dead killed during the Korean War. They signed a joint statement reaffirming North Korea's commitment to giving up nuclear weapons.
Pompeo, Trump's point person for all things North Korea, conceded Washington and Pyongyang "still have a long ways to go".
The United States sent a DNA sample of the South Korean soldier's remains earlier this year that confirmed him as Yun. "We will be ready".
Pyongyang has consistently demanded a more gradual process based on developing mutual trust.