Asked if he would meet with Mr. Kim, Mr. Trump said, “We’ll see what happens.”

The cautious American reaction partly reflected a history of suspicion toward the motives of North Korea, which has remained an enemy of the United States since the 1950-53 Korean War.

A pattern of what American officials have described as North Korean duplicity in previous talks has repeated itself, with modest variations, during the Clinton, George W. Bush, and Obama administrations. In January 2009, Defense Secretary Robert M. Gates, who had served under both Mr. Bush and Mr. Obama, summarized the American skepticism about North Korea with an oft-repeated line: “I’m tired of buying the same horse twice.”

The South Korean statement said the two Koreas would begin working-level discussions to prepare for the summit meeting, to be held in the Peace House, a South Korean building in Panmunjom, the so-called truce village that straddles the border. Before Mr. Kim and Mr. Moon meet, the countries will install, for the first time, a hotline by which the leaders can talk on the phone directly, the statement said.

The statement gave no indication that North Korea would start dismantling nuclear or missile programs anytime soon. Nonetheless, the reported agreements represented major progress in Mr. Moon’s efforts to improve relations with North Korea. Those efforts advanced considerably during the recent Winter Olympics in Pyeongchang, South Korea, to which Mr. Kim sent athletes, entertainers and political delegations that included his sister.

The top South Korean envoys who returned from North Korea on Tuesday — Mr. Moon’s national security adviser, Chung Eui-yong, and the director of the South’s National Intelligence Service, Suh Hoon — are expected to be dispatched to Washington this week to brief the Trump administration on their discussions with Mr. Kim.

Mr. Chung told reporters in Seoul, the South Korean capital, that Mr. Kim had been unexpectedly flexible. He said the delegation had expected him to insist that the South and the United States not hold their annual joint military exercises, which were suspended for the Olympics.