Joint U.S. -North Korean military search teams recovered 229 sets of American remains from North Korea between 1996 and 2005.
However, officials say it could be months before excavations can begin and years until the remains are identified.
"Rocket Man" is a nickname Trump used to mock Kim previous year when the two traded insults over North Korea's testing of nuclear weapons and ballistic missiles targeted at the US.
Foreign ministry spokesman Noh Kyu-duk said, "The government hopes the agreements reached between leaders at the North Korea-U.S. summit, including the repatriation of Korean War remains, will be swiftly carried out as a means to fulfill the goal of complete denuclearization and establishment of a peace system on the Korean Peninsula".
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Since the summit in Singapore, there has been increasing scrutiny of the agreement reached by Trump and Kim, with some experts suggesting that North Korea did not really intend to give up its nuclear weapons.
At high-level talks last week, North Korea and the US confirmed they are still poles apart on the next steps to be taken in the process.
The mystery was resolved Tuesday when the official news agency KCNA put out no fewer than four reports on his trip to far-flung Samjiyon county, on the border with China, a lot of them far more detailed than its usual accounts of his "field guidance" visits. The discussions about returning Americans soldiers killed nearly 70 years ago was perhaps the most tangible outcome from Pompeo's trip, which analysts expected would at least secure the release of some remains.
Trump also announced that the US would suspend "war games" with South Korea.
The White House felt that Secretary of State Mike Pompeo's visit last week to Pyongyang, North Korea went poorly, a source told CNN. Reports at press time indicate that those coffins remain on the border but have not been filled with the promised remains.
The Pentagon confirmed this following President Trump's declaration at a speech that week that "we got back our great fallen heroes, the remains sent back today, already 200 got sent back".