"Speed's not that important to me", he said as he began talks with Mr Kim at the Sofitel Legend Metropole hotel on Wednesday.
"At this stage we still don't know what concessions Kim will make or what Donald Trump will do".
Now the two state leaders will have a working dinner.
Speaking to a rally with supporters in late September, Trump took his enthusiasm for his detente Kim to new heights, declaring at a rally with supporters that "we fell in love" after exchanging letters.
"The two men will sit down for about six hours today".
Trump said Cohen "lied a lot" during Congressional testimony in Washington on Wednesday, though he had told the truth when he said there had been "no collusion" with Russian Federation.
We had a very successful first summit.
"It would be a great honor and one that indicates that things on the Korean Peninsula are going in a very favorable direction". "I very much appreciate no testing of nuclear rockets, missiles, any of it", Trump said, adding that he had "great respect" for Kim.More news: Bian: Not vaccinating your child should be illegal
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Little progress has been made on that and other issues since the first Trump-Kim summit in Singapore past year.
Moon has been active in efforts to end confrontation on the Korean peninsula, meeting Kim three times previous year and trying to facilitate his nuclear negotiations with the United States. "They will be met with fire and fury like the world has never seen", Trump says at an event on the topic of opioid usage in the U.S. on August 8, 2017.
The North maintains total control over every aspect of Kim's domestic appearances, but he even answered a shouted question from a foreign reporter - believed to be unprecedented - saying he was "not pessimistic" about the outcome.
The turmoil in Washington has escalated concerns that Trump, eager for an agreement, would give Kim too much and get too little in return.
The two sides have discussed partial denuclearization measures, such as allowing inspectors to observe the dismantlement of North Korea's Yongbyon nuclear reactor, USA and South Korean officials say.
The two leaders had seemed to find a point of agreement when Kim, who fielded questions from American journalists for the first time, was asked if the US may open a liaison office in North Korea.
"The yardstick for assessing. results is not whether Trump proclaims himself 'happy, ' but whether the outcome takes North Korea measurably closer to revealing and dismantling its nuclear and ballistic missile programs".
"If I'm not willing to do that, I won't be here right now", Mr Kim told reporters through an interpreter, when asked if he was ready to give up his nuclear weapons.
North Korea had offered to "permanently and completely dismantle all the nuclear production facilities" at its main complex in Yongbyon if the U.S. dropped sanctions "that hamper the civilian economy and the livelihood of our people", Ri said.
"North Korea would be the same, and very quickly, if it would denuclearize", Trump said on Twitter ahead of the meeting.