The official media has reported that the two leaders plan to meet, but has offered few specifics, including where and when.
A "changed era" had come about, it said.
In a statement released by the state news agency KCNA, it said the two leaders would discuss a "permanent and durable peace-keeping mechanism" for the Korean Peninsula, the "denuclearisation" of the Korean peninsula and other issues of mutual concern.
Washington is demanding the complete, verifiable, and irreversible denuclearization of the North, while Pyongyang has so far only made public pledges of its commitment to the denuclearization of the peninsula - a term open to wide interpretation - while seeking security guarantees. The timing is crucial at this juncture and "it is very important that we defuse little difficulties surrounding the U.S-North Korea summit", Moon said.
A Trump administration official, speaking on condition of anonymity, said the U.S. side was entering the talks with a sense of optimism and an equal dose of scepticism given North Korea's long history of developing nuclear weapons.
The US president has said he hasn't prepped much for these talks.
Trump arrived a few hours after Kim on Air Force One, directly from a tumultuous G7 summit in Quebec, which ended in a bitter personal exchange between the USA president and the host, the Canadian prime minister, Justin Trudeau.
Asked by a reporter how he felt about the summit, Trump said: "Very good".
A diplomatic source in Singapore on Monday told NK News, however, not to "put a great significance" on the working-level meetings.
The jet landed at the airport Sunday afternoon amid huge security precautions on the city-state island.
Ethiopia's Prime Minister Abiy Ahmed Ali arrived on Saturday night on a two-day visit to Egypt where he will hold talks with President Abdel Fatah al-Sisi on bilateral relations.
In October, Trump praised the USA relationship with Singapore and called Loong a "wonderful and loyal partner".
Kim is set to meet Tuesday with Trump in what's shaping up to be one of the most unusual summits in modern history.
Monday's session was held for 2 hours, with the US Ambassador to the Philippines, Sung Kim, taking part.
Past year it carried out by far its most powerful nuclear test to date and launched missiles capable of reaching the USA mainland, while Trump threatened the North with "fire and fury" and Kim dubbed him a "mentally deranged U.S. dotard".
Trump's high-stakes meeting with Kim is scheduled for 9 a.m. local time Tuesday, or 9 p.m. ET. He described it as a "get to know you plus" meeting.
Although gaps remain over what denuclearisation would entail, Trump sounded a positive note in a lunch meeting with Singapore's Prime Minister Lee Hsien Loong.
Trump said last week that Rodman had not been invited to Singapore.
The US team met North Korean Vice Foreign Minister Choe Son Hui and other officials. There was also no sign of his sister, Kim Yo Jong, who has accompanied him to Singapore.