The White House said the daylong summit would also include a working lunch and a larger meeting involving aides to both leaders. The US president was accompanied by Secretary of State Mike Pompeo, National Security Adviser John Bolton, White House Press Secretary Sarah Sanders and White House Chief of Staff John Kelly.
"We anticipate that they will come to their logical conclusion even more quickly than we anticipated", Mr Pompeo said.
Trump and Kim are set to meet on Tuesday at the Capella Hotel on Sentosa for talks which could see North Korea dismantle its nuclear arsenal in return for economic help and security guarantees.
"If diplomacy does not move in the right direction ... those measures will increase".
North Korean leaders, who have been subject to some of the world's toughest ever sanctions, have long wanted to meet with the US president, in part because of the worldwide legitimacy such a summit would provide.
Singaporean Prime Minister Lee Hsien Loong posted on his official Twitter account a picture of him shaking hands with Kim upon his arrival.
Mr Trump had earlier tweeted about the "excitement in the air".
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Despite the initial high stakes of a meeting meant to rid North Korea of its nuclear weapons, the talks have been portrayed by Trump in recent days more as a get-to-know meeting.
With few sources of information other than the state-run media, gossip and word of mouth, most North Koreans are still largely in the dark about the momentous - and potentially life-changing - events that are about to take place outside of their isolated nation.
Iran said North Korea should be wary of negotiating with Mr Trump following his withdrawal from the landmark 2015 nuclear agreement.
Singapore is one of the few countries that have diplomatic relations with both the United States and North Korea.
"If I think it won't happen, I'm not going to waste my time", he said.
Mr Trump withdrew the USA from the 2015 Iran nuclear deal, which the Obama administration had reached with Iran, Britain, France, Germany, China and Russian Federation. The ensuing recognition as a "normal country" could then allow sanctions relief, and later worldwide aid and investment.
Trump insisted last week that the summit would "not be just a photo op", saying it would help craft a "good relationship" that would lead to a "process" that could result in the "ultimate making of a deal".