The White House has previously said that the meeting is slated for May or early June.
During the meeting, Moon received a telegram from Lee Hee-ho, a former first lady of South Korea, congratulating him for a successful summit meeting with Kim on Friday and wishing him a Nobel Peace Prize.
"All we need to take is peace".
National security adviser John Bolton said Sunday on CBS' "Face the Nation" that a decision by North Korea to release the Americans "would be a demonstration of their sincerity" ahead of the still-unscheduled meeting.
But the USA leader's string of bellicose threats directed at Kim and his country - including his vow to rain "fire and fury" on North Korea and to "totally destroy" the country of 25 million people if it threatened the US or its allies - gave rise to global fears of a bloody and possibly nuclear conflict not seen since the days of the Cold War.
Trump tweeted Monday morning "just asking" his followers whether he should meet the North Korean leaders in the demilitarized zone. "That's very nice, thank you. And maybe it won't", Trump said last week about his upcoming talks. I'm just one person. Carranza Ko said this is a win for President Trump and President Jae-In.
He has also said it remains possible that Mr Trump may even travel to the North Korean capital of Pyongyang for the summit. No matter what the likelihood is for Trump's getting a Nobel, his public image among South Koreans has been improving markedly as the mood on the Korean Peninsula shifts from fears of war to hopes of peace.
The historic summit took place in the border village of Panmunjom, where Korean leaders issued a joint statement on plans to materialize before the end of 2018 the permanent end to war in the peninsula.
Kim met with his South Korean counterpart, President Moon Jae-in, at the DMZ for talks last week.
It wasn't clear whether his enthusiasm was stirred by the South Korean president's suggestion Monday that Trump could take the Nobel Peace Prize if the two Koreas win peace.
"The United States has never been closer to potentially having something happen with respect to the Korean peninsula that can get rid of the nuclear weapons ... and peace and safety for the world", said Trump. Those same skeptics worry holding the meeting at the DMZ will appear conciliatory toward Kim.
To whet Kim's appetite for a deal, Moon on Friday handed Kim a computer thumb drive that contained detailed plans to rebuild the North's decrepit economy if Kim denuclearizes his country, aides to Moon said.
In January, Moon said Trump "deserves big credit for bringing about the inter-Korean talks".
He has been accused of ruthlessly executing scores of aging generals and party elite members, including his own uncle.
Kim also was seen speaking in a straightforward manner.