Though skepticism abounds about Kim's motives, he said over the weekend that he would give up all of his nuclear weapons if the United States promised not to invade his country.
United States and North Korean officials are still negotiating the timing and location of the possible summit.
"Everybody wants us. It has the chance to be a big event", the president said on a bright spring day in Washington, alongside Nigerian President Muhammadu Buhari, whom he'd just met at the White House.
That's what South Korean President Moon Jae-in has suggested the credit-hungry US president receive for his role in the ongoing North Korean nuclear crisis.
There is a "strong possibility" the summit will be held at the site, with some events possibly scheduled on the northern side of the military demarcation line separating the two countries, according to an official with deep knowledge of North Korea's thinking on the matter.
"Kazakhstan as a country that has experience in nuclear disarmament also welcomes the Panmunjom Declaration for the Peace, Prosperity, and Unification on the Korean Peninsula and fully supports the agreement between the two countries on seeking the denuclearization of the peninsula", the Kazakh Foreign Ministry said.
"If it doesn't work out well, that's the way it goes", resolved Trump.
It wasn't clear whether his enthusiasm was stirred by the South Korean president's suggestion Monday that Trump could take the Nobel Peace Prize if the two Koreas win peace.More news: Rams ship WR Austin to Cowboys for sixth-round pick
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"All we need to take is peace".
If they were genuinely grateful to Trump, they were also seen as stoking the ego of the impulsive USA leader so that he would continue to support South Korea's efforts to resolve the North Korean crisis through dialogue.
Trump has also been eager to take credit for Kim's shift in behavior as the inter-Korean summit has fanned speculation that he could receive the Nobel Peace Prize.
Previous winners of the Nobel Peace Prize have included Mr Trump's predecessor Barack Obama, Pakistani activist Malala Yousafzai and Nelson Mandela. "And maybe it won't", Trump said last week about his upcoming talks.
Kim and Moon have pledged to seek a formal end to the Korean War by year's end and to rid the Korean Peninsula of nuclear weapons.
"I think it will probably be in the Joint Security Area at Panmunjom in the demilitarised zone".
Speaking to reporters Monday, Trump said he was enthusiastic about the idea of holding the summit in the DMZ.
Trump has accepted an invitation for a meeting with Kim. Lindsey Graham, R-S.C., a Trump ally who is sometimes critical of the president, said that if Trump "can lead us to ending the Korean War" while "getting North Korea to give up their nuclear program" in a verifiable way, then he "deserves the Nobel Peace Prize and then some".