Mr Moon proposed the compromise at a press briefing in Seoul after a three-day visit to North Korea.
After two days of talks, Kim pledged to close a missile testing site that was crucial to North Korea's development of intercontinental ballistic missiles, and offered to shut the country's only known nuclear complex in exchanged for unspecified "reciprocal measures" from the US.
South Korean President Moon Jae-in says North Korean leader Kim Jong Un hopes to meet again with United States President Donald Trump.
Moon said Kim was also open to inspection of a nuclear test site in the northwest town of Punggye-ri, which he called the North's sole existing facility for underground detonations.
North Korea has been demanding a declaration formally ending the Korean War, which was stopped in 1953 by a ceasefire, but neither leader mentioned it as they read the joint statement.
Both Mr Trump, who has repeatedly spoken of his good relationship with Kim, and the North Korean leader have expressed a desire to meet again, but there are worries among observers about whether Mr Kim is as committed to denuclearisation as he claims.
"Such measures should be taken in a balanced manner between North Korea and the United States", he said.
Other than that, however, there are scant details about what Pyongyang's demands are, or what concrete steps it would take to fulfill Kim's commitment to Trump in June.
The January 2021 completion date was the most specific deadline set in what is expected to be a long process of trying to get the North to end its nuclear program, which may threaten United States allies South Korea and Japan as well as the USA homeland.
Asked why this detail was not in the document signed by Moon and Kim, Nauert replied: "We have had conversations. with the government of North Korea and that is our mutual understanding; that is also the understanding between (South) Korea and North Korea". "I feel we are not making sufficient progress with respect to the denuclearization of the Korean Peninsula", Trump tweeted in late August after cancelling a planned trip to North Korea for Pompeo.
Mr Pompeo cited North Korean leader Kim Jong-un's decision to "complete the previously announced dismantlement of the Tongchang-ri site in the presence of United States and worldwide inspectors as a step toward the final fully verified denuclearisation of North Korea".
USA officials are in touch with the North Korean regime, Pompeo said on Wednesday.
Only hours after standing with Kim on the peak of a volcano that's at the heart of Kim dynasty propaganda, Moon told reporters in Seoul that he will be carrying a private message from Kim to Trump about the nuclear standoff when he meets the USA president in NY next week on the sidelines of the U.N. General Assembly session.
"Chairman Kim and I share the history of having held hands like lovers and crossed the Military Demarcation Line together twice", Moon said during a toast at a banquet Tuesday evening.
An end-of-war declaration would not affect the presence of USA troops and the UNC in the South, Moon said, adding that Kim shared his view. The 1950-53 war still technically continues because it ended with a cease-fire, not a peace treaty.
A better-than-expected outcome of the summit between the two Koreas immediately kick-started stalled negotiations between Washington and Pyongyang, boosting President Donald Trump's high-stakes push to get the North to give up its nuclear weapons by the end of his first term in office.
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