After the summit meeting, Mr. Trump characterized the annual exercises as “very provocative” — a description that aligns with North Korea’s views and sharply deviates from his own Defense Department. The Pentagon has long insisted that the exercises are not meant to provoke North Korea; rather, military officials said, they underline American commitment to its allies in the region and seek to ensure that South Korea, in particular, will be able to defend itself.

Defense Department officials said on Friday that they expected an announcement, most likely next week, that the Ulchi Freedom Guardian exercise will either be canceled or be scaled back sharply. Defense Secretary Jim Mattis and his South Korea counterpart, Song Young-moo, discussed canceling the exercises during a telephone call on Thursday, said a Defense Department official who spoke on the condition of anonymity because the announcement had not yet been made.

Officials in Seoul have also strongly indicated that the exercise will be suspended, with President Moon Jae-in agreeing to be “flexible” as long as North Korea stays engaged in dialogue and starts moving toward dismantling its nuclear weapons program. That comment was widely taken as endorsing Mr. Trump’s decision to end the exercises on the same conditions.

Publicly, the Pentagon has said only that Mr. Mattis reaffirmed the “ironclad” alliance with South Korea. Pentagon officials said Mr. Mattis made similar reassurances by phone on Thursday to Japan’s defense minister, Itsunori Onodera.

Mr. Pompeo was returning to Washington on Friday after nearly a week of clarifying Mr. Trump’s intentions after the Singapore meeting. In Seoul on Thursday, for example, Mr. Pompeo played down the president’s claim that North Korea no longer represented a nuclear threat.

“When he talked about the reduction in nuclear threat,” Mr. Pompeo said, “it was with eyes wide open.” He added that “it could be the case that our effort will not work, but we are determined to set the conditions.”