Moon, who met with senior aides at the presidential Blue House in Seoul on Monday, also said he "evaluates positively" a statement from Kim last week that suggested the North Korean leader was open to a third summit with Trump, News 1 reported.
"Now is the time to fully prepare for a summit of North and South," Moon told gathered officials. Depending on North Korea's circumstance, North and South Korea ought to meet for "concrete and substantive" discussions that would achieve progress beyond the two U.S.-North Korea summits, he added.
Trump most recently met with Kim in Vietnam, but the talks ended without a deal.
Ko Min-jung, deputy spokeswoman at the Blue House, said Moon is concerned with how to "make peace on the Korean Peninsula."
The South Korean president is expected to leave for a tour of Central Asia on Tuesday, but his national security adviser, Chung Eui-yong, is not expected to accompany him.
Chung acted as a special envoy to North Korea in 2018, ahead of summits between leaders.
On Monday Moon also told South Korean military generals they should be prepared to "meet rapid changes on the Korean Peninsula," and that the country has "not completely arrived at peace."
"As we solidify the U.S.-South Korea alliance and play the role of safeguarding the peace of Northeast Asia, we need a strong military," Moon said, according to Yonhap.
Moon has supported civic exchange with the North. New South Korean Unification Minister Kim Yeon-chul is to meet with North Koreans to discuss arrangements for video reunions for separated families in the North and South.
Newsis reported Monday there are currently 55,000 South Koreans who have identified themselves as separated families. Their average age is 81.5, according to the report.