Asked during the unprecedented question-and-answer session if he was willing to allow the U.S.to open an office in Pyongyang, Kim said through a translator, "I think that is something which is welcomable".
"If I weren't willing to do that, I wouldn't be here right now", Kim said in response to a question whether he's ready to denuclearize.
Earlier, both Trump and Kim had expressed hope for progress on improving relations and on the key issue of denuclearization, in their talks in the Vietnamese capital, Hanoi, their second summit in eight months.
Possible outcomes could include a peace declaration for the Korean War that the North could use to eventually push for the reduction of USA troops in South Korea, or sanctions relief that could allow Pyongyang to pursue lucrative economic projects with the South. "We're having very, very productive discussions".
In May, Kim released three U.S. citizens detained in North Korea, which Trump hailed as a "positive gesture of goodwill".
The main negotiation part of the summit lasted more than two hours.
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Mark Meadows was the opposite, doubling down on the congressional tradition of narcissistic grandstanding and weird distractions. Afterward, Patton told PBS Newshour reporter, Yamiche Alcindor , that to her "today was not about the color of my skin".
"At this stage we still don't know what concessions Kim will make or what Donald Trump will do. From what I feel right now, I do have a feeling that good results will come out".
Kim noted the attention the summit is garnering, likening it to a fantasy movie while saying he will do his best to "bring a good result". "I would not say I'm pessimistic".
Washington has demanded North Korea's complete, verifiable and irreversible denuclearization before sanctions can be lifted, a position Pyongyang has denounced as "gangster like".
President Trump reiterated NKorea's potential, if a deal can be done, saying the isolated country could be an "economic powerhouse". A political declaration to end the war and perhaps diplomatic liaison offices would enable Kim to harness changes to North Korean society, like a rapid expansion of markets, argues Bob Carlin.
The U.S. president addressed reporters' questions, responding again with an emphasis that there is "No rush".
It is not clear what the joint agreement will include.
Asked by a reporter if they would declare a formal end to the Korean War, which was halted with an armistice in 1953, Trump replied: "We'll see".
Trump also said he believed the autocrat's claim that he had nothing to do with the 2017 death of Otto Warmbier, a American college student who died after being held in a North Korean prison.
North Korean state media have praised Mr Kim for making the 4,000km (2,500-mile) trip, with state paper Rodong Sinmun dedicating four out of its six pages to it.