Trump, who has hinted at imminent news about three Americans detained in North Korea, once more sounded an upbeat note regarding their fate.
South Korea's national security adviser, Chung Eui-yong, is now in Washington for talks with his USA counterpart John Bolton.
During his presidential campaign, Trump complained that South Korea does not do enough to financially support the American military commitment.
Another sign of the rapprochement will come next week, when a team from the UN's International Civil Aviation Organization will travel to North Korea to discuss a proposal to start an air route between the Pyongyang and Incheon, South Korea, airspace regions, according to Anthony Philbin, the agency's communications chief.
Speaking about the summit, Mr Trump said: "We now have a date and we have a location".
North Korea went forward in time by 30 minutes on Saturday, state media said, to match its clocks with those of the South following last week's inter-Korean summit.
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The US president suggested earlier this week that the Demilitarized Zone between the two Koreas - scene of the historic Kim-Moon talks - could also be an appropriate venue for his own meeting with Kim.
Later speaking to reporters at the Andrews Joint Air Force base, Trump denied reports that he was planning to withdraw United States troops from S Korea. Trump has expressed frustration that South Korea does not contribute enough to fund US forces that bolster its defenses.
In his remarks to reporters Friday, Trump said his administration is in "constant contact" with North Korean leadership. Under the current agreement that expires at the end of 2018, the South provides about $830 million per year.
File images of Kim Jong-Un and Donald Trump.
"A lot of things have already happened with respect to the hostages and I think you're going to see very good things", Trump said from the White House's South Lawn.
President Trump did not share details on the time and place for the landmark meeting with Mr Kim. "As I said yesterday, stay tuned", Trump said, referring to an earlier tweet on the issue.