Kim was heard telling Trump through a translator: "I think the entire world is watching this moment".
Trump did not refer to the document as a treaty or agreement.
He gave no further details.
Kim, dressed in a dark suit, and Trump, sporting a red tie, talked in a corner of the hotel corridor for a few minutes before they walked along the passage into the meeting room and sat down in front of journalists. "A tremendous success. We will have a great relationship".
Kim says through an interpreter that it "was not easy to get here" and that there "were obstacles but we overcame them to be here". "Many people in the world will think of this as a scene from a fantasy...science fiction movie".
Cheering crowds greeted Kim during his stroll, and cameras from North Korea's KCNA documented his every move.
"We may be able to recoup a little bit of that but I think it is a cost that we're willing to pay and it's our contribution to an global endeavour, which is in our profound interests", said Lee during a press conference held at the summit's worldwide Media Centre.
Reporter: "The man you met today, Kim Jong-un, has killed family members, has starved his own people, is responsible for the death of Otto Warmbier".
Mr. Trump was accompanied in a second meeting by a group of senior officials that included national security adviser John Bolton, who angered North Korea with comments suggesting it should follow the "Libya model".
The leaders will meet on Tuesday on the resort island of Sentosa and talks are expected to centre on ending the North's nuclear weapons and missile programmes in return for diplomatic and economic incentives.
Trump said Saturday that he believes the North Korean leader will "surprise on the upside".
Both North Korea and the US have nonetheless sought to set expectations high.
"Wide-ranging and profound views on the issue of establishing new DPRK-US relations, the issue of building a permanent and durable peace-keeping mechanism on the Korean Peninsula, the issue of actualizing the denuclearization of the Korean peninsula and other issues of mutual concern" will be discussed, state-run KCNA reported early Monday.
More officials took part in the working lunch, with the additional participants on the US side, including White House Press Secretary Sarah Sanders, National Security Council Senior Director for Asia Matt Pottinger, and senior diplomat Sung Kim, who has led substantive talks with Pyongyang into the summit.