"We are hopeful these discussions on July 12 will lead to further discussions and negotiations directly with the North Koreans by which we can actually get down to the detailed planning", McKeague said.
At Seoul's invitation, Pyongyang sent athletes and senior officials to the South's Winter Games in February and South Korean basketball players visited Pyongyang last week. Reports say North Korea is now asking for higher-level talks for Sunday. "Great progress being made", Trump said in a tweet that also included a copy of the correspondence from Kim.
In another development on Thursday, the United States accused North Korea of violating a UN sanctions cap on refined oil products.
According to President Trump, returning the remains of US troops was agreed upon during the Singapore summit.
"Rocket Man" is a nickname Trump used to mock Kim past year when the two traded insults over North Korea's testing of nuclear weapons and ballistic missiles targeted at the US.
More than 36,000 USA troops died in the three-year war, which ended in an armistice instead of a peace treaty, leaving the peninsula divided by one of the world's most heavily fortified borders.
President Trump posted Kim's personal letter on social media Thursday, saying he just received a "very nice note from Chairman Kim of North Korea".
"If those requests were "gangster-like" then the world is a gangster, because there was a unanimous decision at the U.N. Security Council about what needs to be achieved", Pompeo replied.
US officials and experts have said the shift in attitude was to be expected as the North Koreans are known for brinkmanship and trying to up the ante ahead of tough negotiations over their nuclear missile program.
The Department of Defense estimates that North Korea is holding about 200 sets of remains from some 5,300 American military personnel believed missing in the country.
But North Korea's delegation failed show up.
The White House had earlier said Mr Pompeo would meet with Mr Kim, but he did not turn up and he only met with Kim Yong Chol, the North Korean leader's powerful right-hand man. The news agency noted that the request followed North Koreans not showing up to the meeting with USA representatives.
There is speculation that North Korea may also want payment for the return of the remains, our correspondent adds.