Mr. Pompeo raised the issue of the detained Americans, according to a senior official. But much of his one-day visit was devoted to logistical issues, like the venue and date for a meeting, which is expected in late May or early June. The lack of involvement by the White House or the State Department, another official said, has limited the amount of substantive preparation for the meeting with Mr. Kim.

General McMaster’s hawkish successor, John R. Bolton, is another wild card. Two weeks before he was recruited as national security adviser, he said a meeting between Mr. Trump and Mr. Kim was useful only because it would inevitably fail, and then the United States could move swiftly on to the next phase — presumably a military confrontation.

“It could be a long and unproductive meeting, or it could be a short and unproductive meeting,” he said on Fox News.

Since entering the White House, however, Mr. Bolton has stuck to a traditional definition of his job, brokering proposals to present to Mr. Trump, officials said. Even among officials who worry about war, there is sympathy for his view that “failing quickly” would be valuable. The United States, they said, should flush out Mr. Kim’s intentions before he has another six months or a year to master intercontinental ballistic missiles.

Mr. Bolton’s presence has also not stopped Mr. Trump from praising Mr. Kim and voicing optimism — even excitement — about their looming encounter. On Friday, he tweeted, “North Korea has agreed to suspend all Nuclear Tests and close up a major test site. This is very good news for North Korea and the World — big progress! Look forward to our Summit.”

The administration viewed Mr. Kim’s statement about halting nuclear tests as more intriguing than his acceptance of a continuing American troop presence because he made it to his own people. Still, they noted his unwillingness to rule out short- and medium-range missiles, which they said could divide Japan from the United States.

Officials also acknowledged the challenge of staying in sync with Mr. Moon, who is acting as a mediator between the United States and North Korea and who is deeply invested in ending years of estrangement between the North and South.