Mr. Moon and Mr. Trump also agreed to “push for maximum pressure and sanctions against North Korea” at the United Nations Security Council, Mr. Park said.

But Mr. Moon has argued that sanctions and pressure alone have failed to stop the North’s advances in nuclear and missile technology. And while Mr. Trump has threatened North Korea with “fire and fury,” Mr. Moon has said there must be a peaceful solution because South Koreans, not Americans, would bear the brunt of war.

He called the North Korean nuclear test on Sunday “disappointing and infuriating.”

Lee Seong-hyon, an analyst at the Sejong Institute near Seoul, said: “Trump considers Moon naïve for insisting on dialogue with North Korea when it keeps conducting missile and nuclear tests. Trump is asking Moon, ‘Are you with us or not?’”

Mr. Trump’s threat to cancel the trade agreement squeezes South Korea on its economy, which is already suffering in part because of South Korean cooperation with the United States.

After North Korea tested its first intercontinental ballistic missile in July, Mr. Moon responded by authorizing an American missile-defense system that the Americans had been pushing for and that he had resisted. But authorizing the missile defense system angered China, which retaliated by cutting back on South Korean consumer products, forcing the South Korean carmaker Hyundai to temporarily shut down assembly lines in China.

Mr. Trump is not the first American leader to be skeptical of a progressive South Korean president’s approach to the North. Since North Korea was first discovered to be developing nuclear weapons in the 1990s, the allies have not always been on the same page.

Washington feared that liberal South Korean presidents’ preference for dialogue and openness were helping North Korea buy time and secure funds for its nuclear weapons program. South Korean progressives argued that dialogue with the North slowed down its nuclear program and even halted it, at least for a while.

