For the moment, Mr Moon has maintained an appearance as a neutral middleman who can bridge the gap between Mr Trump and Mr Kim, two reactive leaders who create a high risk of miscalculation.
Hooker was also a part of a US delegation led by Ivanka Trump, daughter of the president and his senior adviser, that attended the closing ceremony of the PyeongChang Winter Olympics at the end of February.
Any attendance by Mr Moon would be dependent on the progress of pre-summit talks between the U.S. and North Korean leaders, said the Blue House.
General Kim Yong Chol landed at Beijing airport on Tuesday and will journey on to NY the following day after talks with Chinese officials, according to South Korea's Yonhap news agency, which cited diplomatic sources.
The president made the comments in a tweet on Sunday as United States officials arrived in North Korea and met their counterparts at the border village of Panmunjom. Kim is a former American top nuclear envoy and ambassador to South Korea. We're looking at June 12 in Singapore.
China, South Korea and the U.S. all back denuclearisation of the Korean Peninsula, but differ on how to make that happen.More news: Nipah kills one more in Kerala; public meetings banned, exams put off
But it has been thrown into uncertainty, after Mr Trump said he would no longer be attending.
After pulling the United States out of the Iran nuclear agreement and aggravating allies with trade tariffs on steel and aluminum, a potential success with Kim would have created an opportunity for the former NY businessman and his supporters to tout his unconventional style as beneficial to the world.
"He... expressed his intention to put an end to the history of war and confrontation through the success of the North-US summit and to cooperate for peace and prosperity", Moon told reporters in Seoul on Sunday.
Pictures from the summit showed Mr Moon and Mr Kim shaking hands as well as embracing, and revealed Mr Kim brought along his sister, Kim Yo Jung.
"Chairman Kim clearly appealed once again that his intent to completely denuclearise the Korean Peninsula is firm", Mr Moon said.
Members of Congress said the devil, as is always the case with North Korea, will be in the details.
When national security adviser John Bolton first raised the "Libya model", he was not referring to the overthrow of Moammar Gaddafi. "He knows what he can concede and what it means and what he cannot concede". -North Korean summit that he sees as the best way to ease animosity that had some fearing a war past year. "And I think unfortunately, the answer is no".More news: Harvey Weinstein arrested on rape, criminal sex act charges
"Ultimately, I remain convinced that he does not want to denuclearize - in fact, he will not denuclearize". "This is an opportunity to find out what, in fact, they might be willing to do and vice versa - and that is the most important step right now", Narang said.
Trump's team believes the "maximum pressure" campaign to strangle North Korea's economy is working, and Kim's regime will have to come to the table eventually, according to a person familiar with the administration's thinking.
Meanwhile US officials are reported to have travelled to the North on Sunday. "I am hopeful that Secretary [of State Mike] Pompeo will brief and engage Congress".
He said on Saturday the USA was still considering the 12 June date for the summit in Singapore.
This weekend President Trump seemed to reverse himself, saying plans for the summit were still moving forward.
A Kim Jong Un impersonator poses for a selfie with tourists at Merlion Park in Singapore on May 27, 2018.More news: Trump says American prisoner in Venezuela has been released
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