The two sides have yet to publicly state they agree on the definition of denuclearization, which could complicate both Tuesday's talks and future rounds.
The US and Singapore delegations enjoyed lobster bisque, beef tenderloin and ice cream.
Pompeo held firm to Trump's position that sanctions will remain in place until North Korea denuclearizes - and said they would even increase if diplomatic discussions did not progress positively.
They also mention the objective of his trip, a "historic first meeting with the U.S. president", describing not only Mr Kim's itinerary but also the fact that he used a Chinese plane to fly to Singapore, rather than his own personal jet.
He has also dined with US Secretary of State Mike Pompeo in NY. He was a former spy chief, reportedly behind assassination attempts in South Korea, the sinking of a South Korean navy ship, and the Sony hack.
"His visit and meeting with Mr Pompeo is a crucial moment in the process of the preparatory negotiations", he said.
If all goes well, the meeting could help pave the way for denuclearisation of the entire Korean Peninsula.
"He is the director of the United Front Department, which handles inter-Korean affairs".
And on Monday, Secretary of State Mike Pompeo appeared to hand the dictator another victory, saying, "We are prepared to take what will be security assurances that are different, unique than America has been willing to provide previously".
Kim Jong-un touring Singapore's Marina Bay Area, on a night-time stroll around some of Singapore's sights.
Kim Yong-chol is a controversial figure, having been linked to the deadly 2010 attack on the Cheonan South Korean warship which killed 46 South Korean sailors. You ever hear that one?
As the United States stepped up last-minute preparations for the much anticipated U.S. "As a factual matter, the North Koreans have never offered to give up their nuclear weapons".
Trump arrived in Singapore on Sunday after a blow-up over trade with other members of the Group of Seven major industrialised nations that cast a cloud over his efforts to score a major foreign policy win in the nuclear talks.
Kim suddenly emerged in his trademark Mao-style suit to walk out to his black limousine, accompanied by his sister, Kim Yo Jong, and aides.
In an op-ed, the Washington Post emphasises how the United States president "has shown an ability to develop seemingly good relationships with some world leaders, and then to undermine them with unexpected words or actions".
He would not say whether that included the possibility of withdrawing US troops from the Korean Peninsula, but said the context of the discussions was "radically different than ever before".
"He is a good interlocutor".
Yet Lewis said that was the exact word used by Kim Jong Il and Kim Il Sung before him, and it does not mean what Pompeo and Bolton would like it to mean.
Rodman is one of the few westerners to have met the North Korean leader on visits to the capital city of Pyongyang.