South Korea's new President, Lee Jae Myung, arrived for talks just hours after Trump criticised the South Korean government in a social media post.
It apparently referred to its handling of investigations related to Lee's conservative predecessor's December attempt to impose martial law.
The remarks cast a dark mood over high-stakes talks for Lee, who took office in June after a snap election that followed Yoon Suk Yeol's impeachment and removal.
Welcoming Lee to the White House's Oval Office, Trump said he was open to negotiating aspects of the US-South Korean trade deal and to meeting Kim.
"I'd like to have a meeting," Trump told reporters.
"I look forward to meeting with Kim Jong-un in the appropriate future."
Lee showered praise on Trump and said he hoped for Trump to engage with North Korea.
South Korea's economy relies heavily on the US, with Washington underwriting its security with troops and nuclear deterrence.
Trump has called Seoul a "money machine" that takes advantage of American military protection.
"Regardless of how this specific incident plays out, Trump has nicely illustrated how unreliable and capricious he is for allies," said Mason Richey, professor at Seoul's Hankuk University.
The two countries are engaged in lower-level negotiations over trade, nuclear power, and military spending.
Several top officials, including the foreign minister, rushed to Washington over the weekend to try to iron out final details.
Trump planned to discuss "North Korea, defence posture, trade barriers, maritime security, American shipbuilding, energy dominance, and more," according to a White House official.
Lee's goal coming into the meeting has been to sidestep some of the thorniest issues, make a good impression, connect personally with Trump, and above all, avoid any unpleasant surprises, analysts said.
As part of his preparations for the summit, Lee told reporters during his flight to Washington that he had read "Trump: The Art of the Deal," the president's 1987 memoir.
Under heavy pressure from Trump's administration, South Korean negotiators secured a last-minute deal last month to avoid the harshest of new US tariffs, but they must still hammer out details of billions of dollars in promised investments in the United States.
"There are many major topics in the security field," Lee's top policy aide, Kim Yong-beom, said last week.
"Our position is that trade was already finalised last time. We hope that specific implementation plans for trade won't be included in the summit at all, or at least should be kept simple if discussed."
Lee, who arrived in Washington on Sunday, will highlight some of South Korea's expected investments when he visits a shipyard in Philadelphia owned by the country's Hanwha Group after the summit.
Cooperation to help the ailing US shipbuilding sector is part of the broad tariff agreement reached between the countries.
Trump is expected to pressure Lee to commit to more spending on defence, including potentially billions of dollars more toward the upkeep of the 28,500 American troops stationed in South Korea.
