But all that was before the vivid images last Friday of Mr. Kim greeting Mr. Moon at the line of demarcation between North and South, and then sitting down with him in the Peace House.

South Korea built the three-story, gray-stone edifice in the border village of Panmunjom to hold meetings with officials from the North. For years, American officials, including Secretary of State Hillary Clinton, have stopped there during their visits to the Demilitarized Zone. But Mr. Trump missed his chance to visit it during his trip to South Korea last November when heavy fog forced his helicopter to turn around shortly before he was to land.

“Some people maybe don’t like the look of that, and some people like it very much,” the president said. “There’s something I like about it, because you’re there, if things work out, there’s a great celebration to be had on the site, not in a third-party country.”

The Demilitarized Zone has other advantages. The Secret Service is used to securing the site for presidents, and security could be an issue in other countries, like Mongolia, while there are questions about Mr. Kim’s ability to travel long distances, given the rickety condition of his aircraft. Singapore is safe and politically inoffensive, but it is still 3,074 miles from Pyongyang.

Mr. Trump first floated the Peace House in a morning tweet that felt like a trial balloon or even an exercise in crowdsourcing.

“Numerous countries are being considered for the MEETING,” the president wrote, “but would Peace House/Freedom House, on the Border of North & South Korea, be a more Representative, Important and Lasting site than a third party country? Just asking!”

Mr. Trump’s enthusiasm for the meeting with Mr. Kim has drawn expressions of concern from former officials in both Republican and Democratic administrations. They fear that North Korea will exploit the president’s hunger for a win to lure the United States into a protracted negotiation that will leave it with at least some of its nuclear weapons but vitiate the sanctions regime that Mr. Trump helped build.