In a joint statement at last week's summit, Kim pledged to work toward the denuclearization of the Korean Peninsula in exchange for US security guarantees.
The president said at the cabinet meeting that "they've already sent back, or are in the process of sending back, the remains".
A more effective way of repatriating remains is to have a joint mission between United States researchers and the North Korean military, Downes said.
President Donald Trump had said on June 12 after his summit with Kim Jong Un that the North Korean leader was "already destroying" a missile site, in addition to committing to "complete denuclearization" of the Korean Peninsula.
US Secretary of State Mike Pompeo "will be meeting with them and talking with them at the earliest possible date" to implement what was agreed in Singapore, spokeswoman Heather Nauert told reporters, without providing further details.
It could signal North Korea is willing to include a moratorium on satellite launches in addition to their missile and nuclear tests, which has been a point of contention in past agreements and derailed negotiations in the past.
US and South Korean forces have been training together for years, and routinely rehearse everything from beach landings to an invasion from the North, or even "decapitation" strikes targeting the North Korean regime. "Detailed negotiations have not begun", he said. "I wouldn't expect that at this point".
U.S. officials expect the remains to be handed over to United Nations Command in South Korea at Osan Air Base near Seoul then transferred to Hickam Air Force Base in Hawaii.
Trump, who has been leading an worldwide drive to press North Korea to abandon development of nuclear missiles capable of reaching the United States, told reporters after the June 12 summit that Kim had pledged to dismantle one of his missile installations. A USA official familiar with the process said the remains could be co-mingled - meaning not separated by individual - and could include people who were not American.
Richard Downes, whose father went missing in the Korean War and is president of the Coalition of Families of Korean & Cold War POW/MIAs, was optimistic the Trump-Kim summit would lead to the return of more remains. Going into the summit, Pyongyang repeatedly rejected unilateral nuclear disarmament. The U.S. has long looked to China to use its influence with North Korea to bring it to negotiations, but the visit comes as ties between Beijing and Washington are being tested by a major trade dispute. With Xi standing behind him, Kim could also believe that he would have a stronger negotiating position vis-a-vis the US.
"The existence and risk of proliferation of weapons-usable fissile material on the Korean Peninsula and the actions and policies of the government of North Korea continue to pose an unusual and extraordinary threat to the national security, foreign policy, and economy of the United States", it said.