Kim, 72, has been Kim Jong Un's most trusted policy adviser since the North Korean leader began a peace offensive with the US and South Korea in January.
They shook hands again; Trump gave Kim a thumbs up.
Watch Trump and Kim's historic handshake in the video player above.
The two will huddle alone for roughly 45 minutes before being joined by aides for a larger meeting and working lunch.
However, Watanabe of the Sasakawa Peace Foundation said it's time for Japan to reach out to North Korea on its own instead of counting on the mercurial Trump to mediate an Abe-Kim summit. "We will all know soon whether or not a real deal, unlike those of the past, can happen!"
While advisers say Trump has been reviewing briefing materials, the president insists his gut instincts will matter most when he gets in the room with Kim.
AFP 2018 Saul Loeb World Follows Every Step of Historic Trump Kim Summit
North Korea's central news agency on Monday proclaimed a "changed era", and promised talks on "the issue of building a permanent and durable peace-keeping mechanism on the Korean peninsula" as well as "the issue of realizing the denuclearization of the Korean peninsula".
If Trump and Kim fail in Singapore "the result may be to enhance North Korean dependency on Seoul and Beijing as safety valves against the prospect of renewal of U.S".
Trump and Kim will participate in a one-on-one meeting, with only translators, followed by an expanded meeting that will include Secretary of State Mike Pompeo, Chief of Staff John Kelly and National Security Advisor John Bolton.
Kim, speaking through an interpreter, said: "There were fetters and wrong practices that at times covered our eyes and ears, but we overcame everything to come this far". This has included backing away from an original demand for North Korea's swift denuclearisation.
Trump spoke to Moon and Japan's Prime Minister Shinzo Abe, both key allies of Washington in the region, to discuss developments ahead of the summit. In some instances, past deals struck by previous USA administrations with North Korea paused - but failed to roll back - North Korea's program, which has only grown over the years.
As the final preparations for the summit were being made, some veteran diplomats warned against an all-or-nothing approach from the U.S. Kim had his own harsh words, calling the president a "mentally deranged US dotard".
Lee added that while the government has been "cost conscious" in its approach to spending for the summit it has also worked to ensure that all operational requirements for the meeting - especially in terms of security - have been met.More news: LeBron James' future with Cleveland Cavaliers in doubt after NBA Finals loss
More news: Trooping The Colour: The Queen at 92 watches as guardsmen march
More news: Trump blasts Canada, calls for Russian Federation to be returned to G8