For his part, Kim, when asked whether he was ready to denuclearize, had said, "If I'm not willing to do that I won't be here right now".
Trump says that he considers it "a good idea", adding that it should happen "both ways".
Kim and Trump sat down again at the Sofitel Legend Metropole Hotel in Hanoi, where the two leaders had a brief one-on-one chat and a dinner for about two hours Wednesday night.
Expectations were raised in the morning when Kim told reporters that he would not have come to Hanoi if he was not intent to draw up a deal on denuclearization, while Trump claimed that they would "ultimately have a deal".
Meanwhile, South Korean President Moon Jae-in plans to offer new proposals for inter-Korean engagement following the summit between President Donald Trump and North Korean leader Kim Jong Un.
Earlier, North Korean leader, Kim Jong-un said he would welcome the opening of a U.S. liaison office in his capital. "I very much appreciate no testing of nuclear rockets, missiles, any of it".
Director of National Intelligence Dan Coats told a congressional panel last month that North Korea "has halted its provocative behavior" by refraining from missile tests and nuclear tests for more than a year.
At former President Ronald Reagan's first meeting with then-Soviet leader Mikhail Gorbachev in Geneva, Switzerland, in November 1985, the two men met alone with only trusted interpreters.More news: Victoria's Secret closing more than 50 stores after poor holiday sales
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Mr Trump earlier said there was "no rush" to reach a deal.
He added: "Chairman Kim if you're smart, you'll take the way out given to you by President Trump if you're actually trying to play the guy, you're gonna regret this effort".
The summit was "pageantry for Trump and brings Kim more credibility on the world stage as a responsible, rational actor", the Stimson Center's David Kim told AFP. "I wouldn't say that I'm pessimistic".
Trump said the United States would be able to inspect some North Korean facilities but he did not go into specifics.
Trump reiterated North Korea's potential, if a deal can be done, saying the isolated country could be an "economic powerhouse".
The White House has given no indication as to what the signing ceremony might involve, although the two sides have held discussions that have included the possibility of a political statement to declare the 1950-53 Korean War over, which some critics say would be premature.
Daniel Russel, who served as the top USA diplomat for East Asia until early in Trump's presidency, said telling Kim that he could take his time was anything but strategically wise.
President Trump is allegedly to leave the summit venue about thirty minutes after that, . before which we may hope for some more quality time between the two leaders.