Kim held a meeting with Singapore Prime Minister Lee Hsien Loong ahead of the planned summit.
"I think the minimum would be a relationship. We were able to complete the preparations for the historic summit, and I would like to thank you for that", Kim, speaking through an interpreter, told Lee.
According to the White House, Trump will then leave the country that same evening.
That modest vision marks a break with the stratospheric goals he set after he agreed on March 8 to this meeting.
Mr Trump has said he has a "good feeling" about Tuesday's much-anticipated summit.
Trump had been scheduled to fly back to Washington on Wednesday morning after spending Tuesday with the North Korean leader in Singapore. However, judging by what American officials were saying, the United States wants North Korea to give up its nuclear weapons completely under worldwide supervision.
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.
Lee has said the summit would cost Singapore about S$20 million ($15 million), more than half of which would go on security.
The president who once described Kim as "Little Rocket Man" and threatened to destroy his country if it did not give up nukes, has not said what might happen if the summit goes the wrong way.
But he said that any past USA president could have gotten a one-on-one meeting with the leader of North Korea.
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It "would instead lay out practical steps with requisite commitments from North Korea that demonstrate that it understands, accepts and is committed to achieving denuclearisation as the United States defines it", said Enos, who is also in Singapore to watch the proceedings.
Plans for the meeting almost foundered after Trump abruptly withdrew in May, citing North Korea's "open hostility" during negotiations.
Pompeo has disputed the notion that the president is moving the goalposts, telling reporters Thursday that Trump "has always understood that this was a process".
Olivia Enos of the US -based Heritage Foundation is urging Trump to not ignore rights concerns.
Few experts believe that Kim, who has conducted several missile tests in recent years, is willing to dismantle his entire nuclear program.
Kim and President Trump arrived in Singapore on Sunday ahead of Tuesday's historic summit. It will be the first ever meeting between a sitting US President and North Korean leader.
What did the secretary of state say?
Kim has been more cryptic and calculating, with his regime vacillating between welcoming talks with the USA, ignoring American overtures, and sharply attacking some of Trump's top officials with insults and threats, said Patrick Cronin, director of the Center for a New America Security's Asia-Pacific security program.
Trump boasted Saturday that he will be able to deduce nearly immediately whether Kim is serious about the prospect of disarmament. Before Monday, North Korea's state media had reported only that Kim and Trump would be meeting at some stage, but had not provided any further details, reported the AP.