"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.
Kim's missive also hinted at the possibility of another summit, saying that the "epochal progress in promoting the DPRK-US relations will bring our next meeting forward".
I firmly believe that the strong will, honest efforts and unique approach of myself and Your Excellency Mr. President aimed at opening up a future between the DPRK and the USA will surely come to fruition.
"We had a productive conversation", the USA top diplomat said, but added, "There remains a great deal of work to do".
The two sides were supposed to meet at 9 a.m.in the border village, but the North Koreans did not show up, according to officials from the government and the United States Forces Korea (USFK).
South Korea's Foreign Ministry said North Korean military officials were now seeking a meeting Sunday on repatriating the remains of the war dead, with Pyongyang wanting to discuss the issue with a US military general.
It's unclear why the North Korean officials were not present.
North Korean officials did not attend a planned meeting with Americans to discuss the return of United States soldiers' remains, officials have said.
Trump agreed to halt bi-annual U.S. military exercises conducted with South Korean forces, while Kim made a vague commitment to denuclearization.
"It's possible that Pompeo proposed July 12 as the date for the meeting and then hastily made it public without confirmation from Pyongyang", a diplomatic source said.
He said he saw a big difference in North Korea's attitudes to talks.
"The road ahead will be hard and challenging and we know that critics will try to minimize the work that we've achieved", he said.
Joint U.S. -North Korean military search teams recovered 229 sets of American remains from North Korea between 1996 and 2005.
Pyongyang may be interested in taking the conversation beyond the scope of US war remains, which would have been the topic of discussion at Panmunjom on Thursday.
However, recovery efforts were suspended when the relationship between both countries worsened, as the North began the advancement of its nuclear programme.