"Rocket man now wants to talk to South Korea for first time," President Trump tweeted, referring to Kim. "Perhaps that is good news, perhaps not — we will see!"
A top South Korean official suggested that the two sides meet as early as next week.
The offer of talks could lead to a temporary relaxation of tensions on the Korean Peninsula. But experts warned that North Korea was most likely borrowing from a well-worn playbook, hoping to win relief from sanctions and buy time to improve its nuclear program without offering any real concessions.
In a New Year's Day speech, Kim said he wanted to ease tensions with the South and was willing to send a delegation to the PyeongChang Olympics, and suggested that the two sides meet to discuss the idea. At the same time, he cautioned the Trump administration that a "nuclear button" was on his desk and his missiles could strike any part of the United States. Trump responded with a threat of his own Tuesday evening.
"Will someone from his depleted and food starved regime please inform him that I too have a Nuclear Button, but it is a much bigger & more powerful one than his, and my Button works!," he wrote on Twitter.
Cho Myoung-gyon, Seoul's unification minister, responded in a televised news conference Tuesday with an offer to meet as soon as Jan. 9 at the shared border village of Panmunjom to discuss cooperation over the Olympics and how to improve overall ties, news agencies reported. Talks, if they took place, would be the first in more than two years.
South Korean President Moon Jae-in favors dialogue to reduce tensions with Pyongyang and sees the Olympics, which will begin Feb. 9, as a "groundbreaking chance" to improve ties and achieve peace. His government is also extremely keen to see the Games go off successfully.
Some analysts cautioned that Kim may be trying to split South Korea from the United States, its ally. Trump and Moon have not been on the best of terms, and Trump has attacked Kim, personally and repeatedly.
Yet Trump did not appear to be disconcerted by Kim's move. He said on Twitter that sanctions and other pressure "are beginning to have a big impact on North Korea," citing the defection of two soldiers from the North across the demilitarized zone into the South in recent weeks.
And while Moon welcomed Kim's address, he stressed that Seoul would have to coordinate the next steps with its allies, according to the Yonhap news agency.
At the State Department, spokeswoman Heather Nauert told reporters Tuesday that "Kim Jong Un may be trying to drive a wedge of some sort between the two nations, between our nation and the Republic of Korea. I can assure you that that will not happen." She said it is up to South Korea if it wants to open talks with the North. But, she added, "we are very skeptical of Kim Jong Un's sincerity in sitting down and having talks."
"I think in the interim there'll be a temporary reduction in tensions, but ultimately this is going to fail, and it's not going to open up some big chasm between Washington and Seoul," said Evan Medeiros, who was the National Security Council's Asia director in the Obama administration and now heads the Eurasia Group's coverage of the Asia-Pacific region.
Kim said Monday that he wanted to improve the "frozen" relations between the two Koreas and would "open our doors to anyone" from the South who sincerely sought national concord and unity.
"We earnestly wish the Olympic Games a success," he said, according to the North's official KCNA news agency. "From this point of view we are willing to dispatch our delegation and adopt other necessary measures; with regard to this matter, the authorities of the North and the South may meet together soon."
After Moon asked his government to move as quickly as possible to bring North Korea to the Olympics, Cho wasted no time in trying to pin down a date.
"We look forward to candidly discussing interests from both sides face to face with North Korea, along with the North's participation in the PyeongChang Winter Olympics," Cho said. "I repeat, the government is open to talking with North Korea, regardless of time, location and form."
Talks could offer an opportunity to dial down tensions on the Korean Peninsula, after a year when war emerged as a real risk.
China's Foreign Ministry welcomed what it called "positive steps" by both sides and said it hoped they would "take advantage of this opportunity and make concrete efforts in improving bilateral ties, and realize denuclearization of the peninsula," according to spokesman Geng Shuang.
But a sticking point has been a planned joint military exercise with the United States, which North Korea sees as preparation for war. Moon has asked Washington for a postponement until after the Olympics, but no agreement has been reached.
Daniel Russel, who served as assistant secretary of state for East Asia under President Barack Obama and is now a senior fellow at the Asia Society Policy Institute, said Kim's aim is to "divide and conquer."
"Kim wants to unwind sanctions and clearly sees President Moon's angst over the Olympics as the weak link in the allied chain," he said. "Diplomacy is always the preferred option, and this opening should be explored carefully. But it would be naive to expect North Korea to negotiate in good faith or consider itself bound by new agreements, given its past record."
Russel said the North Korean leader's behavior fit into a familiar pattern, with the threat of the "nuclear button" on Kim's desk combined with the "enticement" of talks.
"Pyongyang's pattern is to raise tensions to a fever pitch, dangle a conciliatory offer, collect any and all concessions, then rinse and repeat," he said. "The key to disrupting this pattern and compelling North Korea into credible negotiations over its nuclear program — which is the goal of the sanctions — is maintaining unity between the U.S. and South Korea, as well as Japan, China and Russia."
Medeiros argued that North Korea was trying to seek some sanctions relief as international pressure mounts and was "playing for time" as it works on improving its intercontinental ballistic missile program.
It is far from clear what terms Kim might try to attach to talks or what he would bring to the table, said Jonathan Pollack, a senior fellow at the Brookings Institution. Is his government prepared, for example, to announce a suspension of nuclear and missile tests? "Because if they don't, I don't see any realistic possibility of any kind of discussion, and certainly not a negotiation with the North," he said.
Kim's declaration Monday that he had achieved his goal of creating a nuclear deterrent capable of reaching the United States could be the platform for such a suspension, experts say. But they also point out that his statement was effectively a bluff: The North has demonstrated that it could probably deliver a ballistic missile to the United States but not that it could weaponize such a missile with a nuclear warhead, and it would need more tests to demonstrate that ability.
The idea of North Korea attending the Winter Olympics is also not one that meets universal approval.
"The international community was appalled by South Africa's apartheid regime and banned the country from participating in Olympics," said Bruce Klingner, senior research fellow for northeast Asia at the Heritage Foundation. "But in response to North Korea's far more egregious human rights violations — which the United Nations has ruled to be 'crimes against humanity' — the world allows and even encourages North Korea to participate. Why the double standard?"
Klingner said there is a long history of failed attempts to moderate North Korean behavior by entreating Pyongyang to participate in sporting and other cultural events. He recalled the 2000 Sydney Olympics, when the two Koreas marched behind a single non-national flag in return for secret payments to the North and other concessions — a symbolic gesture that failed to improve Pyongyang's behavior.
"Yet with each new attempt there is breathless anticipation that this time will work," he said.
Sen. Lindsey O. Graham (R-S.C.) said the United States should boycott the Olympics if North Korea attends.
"Allowing Kim Jong Un's North Korea to participate in #WinterOlympics would give legitimacy to the most illegitimate regime on the planet," he tweeted. "I'm confident South Korea will reject this absurd overture and fully believe that if North Korea goes to the Winter Olympics, we do not."
Amber Ziye Wang in Beijing contributed to this report.