The highly symbolic trip capped a three-day visit by Moon to North Korea, the first by a South Korean leader in 11 years.
Pyongyang also said it could dismantle its best-known nuclear facility at Yongbyon, if the USA takes "corresponding measures", as Kim and the South Korean president held their third summit this year.
"On the basis of these important commitments, the United States is prepared to engage immediately in negotiations", Pompeo said in a statement.
Earlier today South Korean President Moon Jae-in and North Korean leader Kim Jong-un held a joint press conference in Pyongyang.
Moon said Kim also told him the closure of a nuclear weapons test site at Punggye-ri could be also subject to worldwide verification.
On Thursday, Moon hinted he is aware of Kim's demands of Trump, but he refused to give details saying the matter should be discussed between the U.S. and North Korean leaders.
Washington has demanded concrete action toward denuclearisation, such as a full disclosure of North Korea's nuclear and missile facilities, before agreeing to key goals of Pyongyang, including an easing of global sanctions and an official end to the Korean War.
Pyongyang drew an enthusiastic response from Trump, who tweeted the news coming out of the third inter-Korean summit, saying it was "very exciting!" even though some analysts point out that specific details remain absent and a path to denuclearization is not clear.
Washington meanwhile is still demanding concrete action towards denuclearisation, such as a full disclosure of North Korea's nuclear and missile facilities, before agreeing to key goals of Pyongyang.More news: Amazon considering opening up to 3,000 cashierless stores by 2021: Bbg
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 US allies South Korea and Japan as well as the United States homeland.
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Trump tweeted late Tuesday: 'Kim Jong Un has agreed to allow Nuclear inspections, subject to final negotiations, and to permanently dismantle a test site and launch pad in the presence of worldwide experts.
Asked about Pompeo's statement on Wednesday welcoming plans for the dismantlement of all facilities at Yongbyon in the presence of USA and IAEA inspectors, Nauert said Moon and Kim had talked about inspectors. "We're not out of the woods yet when it comes to North Korea", he said, but "we have some time".
US Secretary of State Mike Pompeo said he had invited his North Korean counterpart Ri Yong-ho to a meeting in NY, since he would be in the city for the United Nations General Assembly meeting.
The South Korean leader also said he'll convey to Trump his and Kim's desire to get a declaration ending the Korean War by the end of this year. "It means North Korea is dismantling its future nuclear capability".
The day after the June 12 Trump-Kim summit, Pompeo said he hoped to achieve "major disarmament" by North Korea by the end of Trump's first term in January 2021.
For its part, Pyongyang - which wants a formal declaration the 1950-53 Korean War is over - has condemned demands for its unilateral disarmament as "gangster-like". "Chairman Kim and I will rebuild our nation, with 80 million people of the North and South holding hand in hand, standing strong".