Mr Kim believes a summit with Mr Trump will be a landmark opportunity to end decades of confrontation, South Korea's President Moon Jae-in said on Sunday following his surprise meeting with the North Korean leader.
So with the summit in Singapore possibly back on, President Moon pledged to call or meet with his North Korean counterpart whenever necessary.
Moon said both he and Kim had "agreed the June 12 North Korea-U.S. summit must be successfully held".
Moon says his meeting with Kim was arranged at Kim's request.
Meanwhile, Trump spoke to reporters about the potential summit from the Oval Office on Saturday night, where he had gathered with lawmakers to celebrate the return of Utah man Joshua Holt, who had been jailed in Venezuela for almost two years and finally freed.
Trump's decision also came while United States and worldwide journalists were in North Korea.
She also said it was up to the Americans whether they would "meet us at a meeting room or encounter us at nuclear-to-nuclear showdown". "If we got that done and if we can be successful in the denuclearization of the Korean peninsula, it would be a great thing for North Korea, it would be a great thing for South Korea, be great for Japan and great for the world - great for the United States, great for China". "That hasn't changed. So, we'll see what happens".
Mr Trump had rattled the region on Thursday by cancelling his meeting with Mr Kim, citing "open hostility" from Pyongyang. During Saturday's inter-Korean summit, the Korean leaders agreed to "positively co-operate with each other as ever to improve (North Korea)-U.S. relations and establish (a) mechanism for permanent and durable peace".
A White House team will leave as scheduled for Singapore this weekend to prepare for the possible summit, a White House spokeswoman said on Saturday.
President Donald Trump says negotiations over a potential summit with the leader of North Korea are "going along very well". Global journalists were present, but the US government can't verify the site's destruction.
Tensions were calmed after Kim extended an olive branch by offering to send a delegation to the Winter Olympics in South Korea, sparking a rapid detente that led to Trump agreeing to hold direct talks with Pyongyang. The US has banned its citizens from entering North Korea, which has a history of detaining US citizens. At that meeting, they declared they would work toward a nuclear-free Korean peninsula and a formal end to the 1950-53 Korean War.
The agency said the two leaders had reached a "satisfactory consensus" and expressed "their stand to make joint efforts for the decentralization of the Korean peninsula".
But within 24 hours he reversed course, saying it could still go ahead after productive talks were held with North Korean officials.
Photos released by South Korea's presidential office showed Moon arriving at the North Korean side of the Panmunjom truce village and shaking hands with Kim's sister, Kim Yo Jong, before sitting down with Kim for their summit.