US officials have laid the groundwork for a second day of meetings if the two men decide they want to continue discussions, according to two sources familiar with the matter.
Trump and Kim are due to meet on June 12 for the first time and the US leader is due to return home the next day, but US officials in Singapore have established a contingency plan for a second day of discussions.
A White House spokesman declined to comment on the matter.
It's not clear if Trump is eager for a two-day summit with Kim, but he has expressed a desire for flexibility in the negotiations and has stressed to aides and US allies that he plans to follow his gut as he negotiates with the North Korean leader.
The US has stressed that it is seeking complete and verifiable denuclearization of North Korea, but a week ahead of the summit, questions remain about whether Kim is truly committed to achieving that aim. US officials emphasize that it is a decision Kim still needs to make.
While leaving open the possibility of a second day of meetings, US officials have been wary of over hyping expectations for what can be accomplished during Trump and Kim's first meeting.
The President joined in that effort last week when he made clear the meeting would be the beginning of a dialogue with North Korea and that it was unlikely he would achieve all his objectives in a first meeting.
"I think it's a 'getting to know you' meeting plus, and that could be a very positive thing," Trump said.
Now, the most some aides are hoping for is a broad declaration from Kim that he's open to getting rid of his weapons, with the work of securing specific commitments from the North Korean regime handed off to aides for a process that experts predict could take years.
On and off summit
The meeting itself is an stunning turn of events after months of antagonism and brinksmanship between the US and North Korea over Pyongyang's nuclear missile program.
That came after an angry rebuke from North Korea in response to suggestions by Trump's national security adviser John Bolton that the country could follow the "Libya model" of denuclearization.
North Korea has not yet reacted to Giuliani's comments.
CNN's Kevin Liptak and Jeff Zeleny contributed reporting.