Skeptics say such agreements would leave in place a significant portion of North Korea's nuclear-tipped missiles while robbing the United States of its negotiating leverage going forward.
Since the first summit in Singapore in June past year, Trump and his administration had claimed they were on the brink of a historic breakthrough in persuading North Korea to disarm, going beyond the agreements forged by former USA presidents.
Earlier the White House said Trump and Kim would have an agreement signing ceremony at the conclusion of the two-day summit which wraps up Thursday.
Both Trump and Kim left the venue of their talks, the French-colonial-era Metropole hotel, without attending a planned lunch together, and returned to their hotels.
"But we couldn't do that", the president said.
Two days later, a statement carried by North Korean state media says: "We cannot have a sound dialogue with a senile man who cant think rationally, only absolute force can work on him".
Speaking to a rally with supporters in late September, Trump took his enthusiasm for his detente Kim to new heights, declaring at a rally with supporters that "we fell in love" after exchanging letters.
Trump also tweeted about Kim, calling the North Korean leader his "friend" and describing the country's potential as "AWESOME".
The leaders are scheduled to meet face-to-face this morning before signing a declaration at 2:00 pm (0700 GMT) today.
Little diplomatic progress was made following the first summit.
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When asked by a member of the White House press pool about his outlook on the summit on Thursday, Kim said: "It's too early to say".
While North Korea's official media said Kim and Trump had made a decision to continue talks, its Vice Foreign Minister Choe Son Hui has struck a more negative tone, telling reporters Kim "might lose his willingness to pursue a deal".
Last year, at the Singapore summit, Trump caught USA ally South Korea off guard by announcing the suspension of major US military exercises with the South. Trump critics said he squandered critical US leverage before the North had taken any concrete steps toward denuclearization.
They have also discussed partial denuclearization measures, such as allowing inspectors to observe the dismantling of North Korea's Yongbyon nuclear reactor, US and South Korean officials say.
The turmoil in Washington has escalated concerns that Trump, eager for an agreement, would give Kim too much and get too little in return.
Asked by a reporter if he was "walking back" on denuclearization demands, Trump said, "No".
Mr Trump and Mr Kim shook hands for reporters before holding talks and having dinner.
But Mr Kim, when asked about denuclearise during an unprecedented question-and-answer session with journalists, said: "If I'm not willing to do that I won't be here right now".
The United Nations and the United States ratcheted up sanctions on North Korea when it conducted repeated nuclear and ballistic missile tests in 2017.
Trump wants Kim to agree to dismantle his nuclear sites in a verifiable way, while Kim is seeking relief from crushing US sanctions hobbling his economy.
Longstanding U.S. policy has insisted that U.S. sanctions on North Korea would not be lifted until that country committed to, if not concluded, complete, verifiable and irreversible denuclearization.