“North Korean offers typically come with caveats and asterisks that need to be examined,” said Daniel R. Russel, a former Asia adviser to President Barack Obama. “We all hope that the multiyear pressure campaign has had an effect, but we shouldn’t prematurely celebrate.”

At the State Department, where some diplomats quietly applauded Mr. Trump’s gamble, there was a fear that more hawkish aides in the White House might throw up further hurdles to the meeting. The White House, they said, has invested more in sanctions and military options than in diplomacy. Officials there have in the past expressed frustration about what they viewed as the Pentagon’s reluctance to provide options for a military strike on the North.

With all the potential traps and internal misgivings, some officials said they believed the chances of a meeting between the two leaders actually happening were less than 50 percent.