SEOUL, South Korea — Secretary of State Mike Pompeo said that in his meeting with Kim Jong-un on Sunday, the leader of North Korea had agreed to allow inspectors into a key nuclear testing site that the North has claimed it blew up, a down payment on the country’s commitment to denuclearize the country.

Mr. Pompeo, whose planned trip to the North in August was canceled by President Trump because of a lack of progress, described his few hours on the ground in Pyongyang, the capital, as a “good trip.”

The two men discussed “the upcoming second summit” between Mr. Trump and Mr. Kim, refining options for the time and place of the next meeting between the two leaders, said Heather Nauert, a State Department spokeswoman. But no date or place was announced.

The office of President Moon Jae-in of South Korea, who was briefed by Mr. Pompeo after his North Korea trip, said that the United States and the North had agreed to hold a new summit meeting between their leaders as early as possible.