SEOUL, South Korea — As President Trump considers a second summit meeting with Kim Jong-un, President Moon Jae-in of South Korea called Tuesday for both leaders to make “bold decisions” that will keep the process of North Korean denuclearization moving forward.

The White House revealed on Monday that Mr. Trump had received a letter from Mr. Kim, the North Korean leader, asking for a second meeting to follow up on their summit talks in June. It was Mr. Kim’s fourth letter to Mr. Trump this year, as the two leaders appeared to cultivate personal ties unthinkable last year, when they exchanged personal insults and nuclear threats.

The White House said it was already looking at scheduling a second summit meeting.

“It was a very warm, very positive letter,” Mr. Trump’s press secretary, Sarah Huckabee Sanders, said during a White House briefing. “The primary purpose of the letter was to request and look to schedule another meeting with the president, which we are open to and are already in the process of coordinating.”

The news was an encouraging signal for Mr. Moon, who has called on Mr. Trump and Mr. Kim to resuscitate the stalled dialogue between their governments. Mr. Moon and many South Koreans hope that Mr. Trump and Mr. Kim will use their personal chemistry to end the North’s nuclear weapons program and bring lasting peace to the Korean Peninsula.