On Friday, Japan's Foreign Minister Taro Kono expressed "understanding" of the US decision to cancel the meeting, saying, "It is meaningless to hold a summit if it does not bring about progress", according to Kyodo News.
"We're looking at June 12 in Singapore. So we'll see what happens".
Kim believes the summit with Trump will be an opportunity to end decades of confrontation, Moon said.
"We got that done and we can be successful in the denuclearization of the Korean peninsula, that would be a great thing for North Korea", Trump added.
Mr. Moon said while Mr. Kim is committed to denuclearisation, North Korea is not convinced it can trust security guarantees from the United States. -North Korea summit on track, said he delivered Trump's "firm will" to end the hostile relationship with North Korea and pursue bilateral economic cooperation.
Mr Trump told reporters that they were still considering Singapore as the venue for their talks.
Coming after a surprise meeting between Kim and South Korean President Moon Jae-in, the move was the latest sign that the on-again, off-again Trump-Kim summit may yet go ahead as planned.
President Donald Trump tweeted Saturday that a New York Times story which indicated that his Cabinet members differ on what he ought to ask for from North Korea was yet another inaccurate piece from the "Failing and Corrupt" publication.
Trump's original decision to abandon the historic summit initially blindsided South Korea, which had been brokering a remarkable detente between Washington and Pyongyang and is desperate to avoid conflict breaking out.
Specifically, Moon and Kim will hold "high-level talks" on Friday, the agency added.
Moon said on Sunday that Kim had requested the meeting - only the fourth ever by leaders of the two countries since the Korean War - and reiterated a commitment to "complete denuclearization" of the Korean Peninsula.
The White House has signaled flexibility over the details of denuclearization, with Press Secretary Sarah Huckabee Sanders saying earlier this month that the administration will follow the "President Trump model". "I urge North Korea and the U.S.to confirm each other's will by sharing their problems with each other and communicating directly".
They agreed to have their top officials meet again June 1 and to set up separate talks between their top generals.
Video and a photo released by South Korea's presidential Blue House on Saturday showed Kim hugging Moon and kissing him on the cheek three times as he saw Moon off after their meeting at Tongilgak, the North's building in the truce village, which lies in the Demilitarized Zone (DMZ) - the 2.5-mile (4 km) wide buffer that runs along the heavily armed military border.