Eyre spoke as U.S. Secretary of State Mike Pompeo headed for his fourth visit this year to North Korea, seeking to break an impasse in denuclearization talks and clear the way for a second summit between Kim and U.S. President Donald Trump.
While North Korea blew up the entrances of the tunnels in May and removed the diagnostic equipment, the shafts could be reopened or new tunnels could be excavated if Kim made a decision to restart underground test blasts, he said.
Cheong Wa Dae said that Pompeo told Moon that had talks with Kim over denuclearization negotiations, including what measures the North Korean leader would take for giving up its nuclear arsenal in a verifiable way - such USA government's inspection of North Korea's nuclear facilities.
But a summit of the leaders of North and South Korea last month has helped to rekindle the peace process, as has the apparent desire of both Kim and Trump to meet again for a second summit.
"The mission is to make sure that we understand what each side is truly trying to achieve", Pompeo said Friday en route to Japan, his first stop on his swing through East Asia.
A United States government plane carrying Pompeo's delegation landed at the Osan Air Base in Pyeongtaek, South Korea, at around 5:15 pm.
"It's a very nice day that promises a good future for both countries", Kim told Pompeo through an interpreter as they sat down for the meal. Moon, who has met twice with Kim, asked Pompeo to make public as much information as he could about the trip.
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South Korean President Moon Jae-in held talks with Pompeo in Seoul on Sunday evening after the top US diplomat met with Kim for more than three hours during a short trip to Pyongyang that was aimed at breaking a gridlock in their nuclear talks.
He will then travel to Seoul for a meeting with South Korean President Moon Jae-in.
Pompeo will also visit Beijing Monday as he concludes his four-nation visit to the region. He is expected to leave for China on Tuesday.
North Korea has dismissed that demand - calling it a "fiction" in a commentary after Pompeo's last visit in July - and the US insistence that North Korea's denuclearization come first, leaving talks at a standstill.
US Secretary of State Mike Pompeo arrives at Sunan International Airport in Pyongyang July 6, 2018.
Washington and Pyongyang have sparred over the exact terms of the vaguely-worded agreement in Singapore, with the USA pushing to maintain sanctions and pressure against the North until its "final, fully verified denuclearisation". "I look forward to seeing Chairman Kim again, in the near future", Trump sad.
Pyongyang and Washington have clashed over the sequence of denuclearization measures and corresponding compensation - a repeat of past nuclear negotiations between the two countries.
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Al Jazeera's Adrian Brow, reporting from Beijing , said China's silence is a "reflection of how sensitive this case is". Since Xi Jinping came to power in 2012, well over a million party officials have been disciplined in some way.
"That could mean the administration offers, and North Korea accepts, a political declaration that ends the Korean War in exchange for a big action towards denuclearization, such as the closing of the Yongbyon nuclear complex".
Until recently, Pyongyang has appeared to be increasing its demands.
Pompeo and Kim also discussed "denuclearization steps that would be taken by North Korea and the issue of attendance by the United States government", as well as "corresponding measures" to be taken by the USA, the statement said.
Japan has been wary of Trump's initiative, fearing it could affect its long-standing security relationship with the U.S. According to Reuters, a commentary issued by North Korea's official news agency KCNA earlier this week said policy experts are "spouting rubbish" when they say North Korea should disclose details of its nuclear program.
The United States wants sanctions on North Korea to remain in place until the country completely dismantles its nuclear program.
Kang said such steps would serve to dispel mistrust between Pyongyang and Washington and speed up negotiations by avoiding the risk of bogging it down over the verification process. "So this is, I think, a good outcome for all of us".
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