So far, the North Koreans have shown no interest in a serious negotiation. For his part, Mr. Tillerson gave no indication of what the administration might be willing to give up if talks began, and Mr. Trump has made clear he would make no concessions. But many inside and outside government have noted there were no major military exercises between the United States and South Korea scheduled until the spring, so the promise of scaling them back could be dangled.

But Mr. Kim would be unlikely to see that as much of a victory and he has rejected any talks that would ultimately require him to disarm.

Speaking at the residence of the United States ambassador to Beijing after a meeting with China’s top leadership, Mr. Tillerson, the former chief executive of Exxon Mobil and a newcomer to diplomacy, was cagey about whether the inquiries yielded anything, or seem likely to.

But hours after he left China, his spokeswoman, Heather Nauert, responding to news reports of Mr. Tillerson’s comments, said in a statement that “despite assurances that the United States is not interested in promoting the collapse of the current regime” or sending American forces into the country, “North Korean officials have shown no indication that they are interested in or ready for talks regarding denuclearization.”

In fact, while the Americans’ outreach was underway, the exchange of public threats between the two countries accelerated. They have included declarations that the North might conduct an atmospheric nuclear test and that it had the right to shoot down American warplanes in international waters.