Trump told reporters on Saturday night that meetings were taking place "as we speak" in an unidentified location to push ahead with the summit, which he abruptly canceled last week due to "open hostility" from North Korea.
"We're doing very well in terms of the summit with North Korea", Trump said.
"We can be successful in the denuclearisation of the Korean Peninsula, that could be a great thing for North Korea, it would be a great thing for South Korea, it would be great for Japan, it would be great for the world, it would be great for the United States, it would be great for China".
The meeting came hours after South Korea expressed relief over revived talks for a summit between Trump and Kim. It allowed Moon to push for a US-North Korean summit that he sees as the best way to ease animosity that had some fearing a war past year.
The leaders of North and South Korea have met in the demilitarised border area between the two countries.
The latest developments came after North Korea said early Friday that it remained willing to meet with Trump despite his decision to scrap plans for the meeting with Kim.
"It's moving along very nicely", Trump said at the White House during a meeting with a U.S. prisoner freed by Venezuela.
"I think people want to see if we can get the meeting and get something done", he said. So we're looking at June 12th in Singapore. "So we'll see what happens".
"However, during the U.S".
Trump's letter to Kim canceling the planned June 12 summit didn't rule out a meeting in the future, said a person familiar with the administration's thinking.
"Chairman Kim requested a meeting without any formality two days ago and I gladly accepted it", Mr. Moon said, noting that the summit was in line with previous agreements for the two leaders to meet more often. North Korea rejected outright calls from U.S. National Security Adviser John Bolton to quickly give up its nuclear weapons before it gets anything in return.
But on Saturday, Mr. Trump said he was still looking at a June 12 date for a summit in Singapore, adding that talks were progressing very well. "I urge North Korea and the U.S.to confirm each other's will by sharing their problems with each other and communicating directly".
Kim reportedly reiterated his pledge to de-nuclearize, but cast doubt on Washington's promise to provide him with security guarantees that the North seeks.