Trump said he wouldn't be meeting Kim, but then added: "I may be speaking with him in a different forum. I will be going, as you know, to South Korea after the summit."
He didn't elaborate.
In a response to questions by AP and six other news agencies, South Korean President Moon Jae In said that Trump's and Kim's "willingness to engage in dialogue has never faded" and that their recent letter exchanges prove that.
Moon, a liberal who met Kim three times last year, has made dialogue with the North - as a means to forging peace on the Korean Peninsula - a centrepiece of his presidency. He has played a central role in facilitating US-North Korean negotiations, even if those efforts have at times been overshadowed by the Trump-Kim talks that he helped broker.
Moon said he doesn't see the Vietnam summit as a failure. He said he thinks the meeting served as a chance for both Washington and Pyongyang to better understand each other's positions and "put everything they want on the negotiating table."
"The success of denuclearisation and the peace process on the Korean Peninsula cannot be determined by a summit or two," Moon said, adding that the discussions in Vietnam will form the basis for future talks. "Both sides clearly understand the necessity for dialogue," he added.
Despite the deadlocked nuclear negotiations, both Trump and Kim have described their personal relationship as good. When asked whether Kim's recent letter included a mention about another summit, Trump said: "Maybe there was."
"But we, you know, at some point, we'll do that," Trump told reporters at the White House. "Getting along very well. He's not doing nuclear testing," he said.
In yet another reminder of North Korea's continued mistrust of the US, its Foreign Ministry said earlier that it won't surrender to US-led sanctions and accused Washington of trying to "bring us to our knees."
Kim has said North Korea will seek a "new way" if the US persists with sanctions and pressure.
Following his setback in Vietnam, Kim travelled to the Russian Far East in April for his first summit with Russian President Vladimir Putin. Kim also hosted Chinese President Xi Jinping in Pyongyang last week for their fifth summit since March last year, and experts say the North's outreach to its traditional allies is aimed at strengthening its leverage with the Trump Administration.
Moon said he views the North's expanding diplomacy with Beijing and Moscow as a positive development in efforts to resolve the nuclear standoff.
"China and Russia have continued to play constructive roles so far to peacefully resolve the Korean Peninsula issue," he said. "I hope that China and Russia will play specific parts in helping the North resume dialogue at an early stage."
Moon didn't elaborate whether US and North Korean officials had face-to-face meetings and if so where they took place. He also didn't clarify who were interlocutors or how close they were in setting up a third Kim-Trump summit.
Trump's top envoy on North Korea, Stephen Biegun, is to visit South Korea today, and some experts said he may use his trip as a chance to meet North Korean officials at a Korean border village. Secretary of State Mike Pompeo said this week that the US was prepared to resume talks with North Korea "at a moment's notice" if the North signalled it wanted discussions about denuclearisation.
Despite a possible restart of negotiations, it's still unclear whether Washington and Pyongyang can eventually achieve agreements that can satisfy both sides. The Vietnam summit fell apart after Trump rejected Kim's calls for major sanctions relief in return for dismantling his main nuclear complex, something that US officials see as a partial denuclearisation step.
Kim has since fired missiles and other weapons into the sea and asked Trump to work out mutually acceptable agreements by the end of this December. US officials maintain sanctions on North Korea would remain in place until North Korea takes significant steps toward nuclear disarmament.
North Korea has long bristled at the significant US military presence in South Korea, and wants assurances it will not be targeted by the US and South Korea. It sees its pursuit of nuclear weapons and ballistic missiles as an insurance policy against military action.
- AP