The State Department declined to comment on Monday. On Sunday night, Stephen E. Biegun, the United States’ special representative for North Korea, told The Times that its account of the ideas being discussed in the administration was “pure speculation” and that his team was “not preparing any new proposal currently.”

Some analysts said any approach must start with the United States and North Korea committing to a common definition of denuclearization. Without an ironclad definition, there is greater risk the North Koreans could back out of an interim deal, as they have done under previous American administrations, they said.

“There’s a myriad of ways that North Korea can pull it back,” said Jung H. Pak, a former C.I.A. analyst who is now a senior fellow at the Brookings Institution.

For months after the meeting in Hanoi, there was no senior-level contact between Washington and Pyongyang, then Mr. Trump and Mr. Kim exchanged letters.

That paved the way for Mr. Trump’s Twitter post from the G20 summit in Japan on Saturday, in which he said he would like to see Mr. Kim during a scheduled visit Sunday to the Demilitarized Zone, on the border between the two Koreas, to shake Mr. Kim’s hand and to “say Hello(?)!”

Mr. Pompeo and other officials scrambled to organize for a potential meeting. Mr. Trump asked Mr. Pompeo to accompany him to the heavily guarded border village of Panmunjom, along with Ivanka Trump and Jared Kushner, his eldest daughter and his son-in-law. Mr. Bolton notably flew to Mongolia instead. “Delighted to be in Ulaanbaatar,” he said on Twitter above a photograph of him smiling with the country’s secretary of state, Davaasuren Damdinsuren.

Though Mr. Pompeo is often aligned with Mr. Bolton on an aggressive approach to national security issues — Mr. Pompeo has also advocated a strike on Iran — the secretary of state is acutely attuned to Mr. Trump’s desires and has tried diplomacy with the North Koreans when commanded by the president.