U.S. President Donald Trump said Monday that a second meeting with North Korean leader Kim Jong Un on denuclearization would happen "in the not-too-distant future," as the South's leader heaped praise on the American president, calling him "the only person who can solve this problem."

Speaking during a televised meeting with South Korean President Moon Jae-in in New York, Trump said there had been "tremendous progress" on the nuclear issue, and that a second summit will be held under a "similar format" as his landmark June meeting with Kim in Singapore, but likely in a different location.

"We'll be having a second summit with Chairman Kim in the not too distant future," Trump said. "I think a lot of progress is being made. I see tremendous enthusiasm on behalf of Chairman Kim for making a deal."