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White House aides had cast Trump's visit to New York in triumphal terms: an opportunity to assert American sovereignty before the multinational body.
Trump hailed "tremendous progress" to halt Pyongyang's nuclear and ballistic missile tests and said that a year later it was a "much different time."
Pompeo told NBC News in an interview that he hoped the meeting could take place before too long.
In June, Trump and North Korean leader Kim Jong-un had met for the first time in Singapore, wherein the latter agreed to repatriate the remains of the US soldiers, who died in the 1950-53 Korean War.
Kim and Moon took to the road for the final day of their summit, hiking to the peak of Mount Paektu, which is considered sacred in the North, their hands clasped and raised in a pose of triumph.
Kim Jong Un stated that he would permanently dismantle North Korea's main nuclear facility only if the United States takes unspecified corresponding measures.
Trump said North Korea saying it would permanently abolish its key missile facilities with witnesses, was very good news.
Kim and Moon have decided to opt for peace, with North's promise of shutting down main missile testing sites.
Moon is on a three-day trip to the North Korean capital for his third summit with Kim this year, hoping to reboot stalled denuclearisation talks between his hosts and the United States.
It was a much warmer welcome than what Kim's late father, Kim Jong Il, arranged for Roh in October 2007.
Negotiations between Trump and Kim have sputtered in recent weeks, raising doubts about whether Kim is truly willing to give up his nuclear arsenal and put pressure on Moon to broker progress.
Trump had told reporters last week that he was expecting a letter from Kim. Sanders said Trump has received the letter, which she described as 'very warm, very positive.'
His comments came after the White House said it was "in the process of coordinating" another summit between Trump and Kim, after Pyongyang proposed a second meeting in a letter delivered to Trump.
The tweet included a quote that said experts believe the missiles were cut from the parade to show Trump North Korea's commitment to denuclearizing.
Kim Jong-un on Wednesday pledged to denuclearise North Korea by the end of Donald Trump's first term in 2021, officials in South Korea said.