SEOUL, South Korea — South Korea is holding its breath for what its president, Moon Jae-in, has repeatedly said could happen by the end of December: Kim Jong-un’s visit to Seoul, which would be a first for a North Korean leader.

When the two Korean leaders met in September in Pyongyang, the North Korean capital, Mr. Kim agreed to visit the South Korean capital “at an early date.” But he has yet to accept Mr. Moon’s invitation to visit by the end of December.

If Mr. Kim doesn’t show up in Seoul this month, it will be a huge letdown for Mr. Moon, who has repeatedly told his people that Mr. Kim promised to try. The South Korean government has been preparing for the visit for weeks. A large artwork showing Mr. Kim and Mr. Moon smiling and shaking hands was even installed last week outside the Blue House, Mr. Moon’s official residence.

But with only three weeks to go, the window appears to be closing for a December visit, and officials have begun sounding less optimistic. And any delay could complicate efforts for a second summit meeting between Mr. Kim and President Trump, though some analysts say that Mr. Kim may see no incentive to meet Mr. Moon before sorting out his differences with Mr. Trump.