In June, without consulting the Pentagon, the president suspended major military exercises with South Korea after meeting with the North’s leader, Kim Jong-un. Two months later, Mr. Trump delivered a rebuke to his defense secretary when Mr. Mattis opened the door to resuming the exercises.

Mr. Trump rejected that and said in a tweet that he saw “no reason at this time to be spending large amounts of money on joint U.S.-South Korea war games.” He added that he could resume them “instantly” if he chose, and that if he did, the exercises would be “far bigger than ever before.”

Vigilant Ace largely involves air maneuvers. Last year’s exercises had more than 200 warplanes from the United States and South Korea. American military officials do not like suspending the exercises because, they say, delaying them can affect military readiness.

Mr. Mattis is in Singapore for a meeting of Southeast Asian defense ministers. While here, his meetings with Asian defense ministers have been shadowed by the continued fallout and speculation ignited by Mr. Trump on Sunday, when he questioned whether the defense chief would remain on the job and said that Mr. Mattis was “sort of a Democrat, if you want to know the truth,” during an interview with CBS’s “60 Minutes.”

Mr. Mattis later told reporters that Mr. Trump had called him to reassure him that he was “100 percent” behind the defense secretary, but in Asia there has been speculation about how long Mr. Mattis will stick around.