The Trump administration is reacting cautiously amid indications North Korea is poised to test another ballistic missile, possibly as soon as this week.
"I hope that does not happen. But if it does, we must bring even more measures to bear on the North Korean regime," U.S. Ambassador Nikki Haley told reporters at the United Nations. "The civilized world must remain united and vigilant against the rogue state's development of a nuclear arsenal. We will never accept a nuclear North Korea."
In response to recent threats, however, U.S. President Donald Trump was more forceful in a tweet Tuesday night:
North Korean Leader Kim Jong Un just stated that the “Nuclear Button is on his desk at all times.” 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!
White House press secretary Sarah Sanders, asked about activity reported north of Pyongyang where North Korea has previously launched long-range missiles, stated that "we're calling on everybody to step up and do more," adding, "We're going to keep every option on the table."
In an interview with VOA contributor Greta Van Susteren on Tuesday, U.S. national security adviser H.R. McMaster expressed concern about Pyongyang's continuing development of weapons of mass destruction, saying, "North Korea has never met a weapon that it has not proliferated."
McMaster expressed hope North Korean leader Kim Jong Un and his generals would recognize "that the continued pursuit of nuclear weapons and missiles is a dead end. And the only way to do that now, short of war, is through coercive economic power — power that mainly rests in the hands of China, but with others as well."
McMaster added that "the trend has been positive in that more and more countries are stopping all trade with North Korea. Vietnam, for example, the Philippines — the list really is quite a long one."
The government in Seoul, meanwhile, on Tuesday accepted an overture from Kim in Pyongyang to hold talks about the North's participating in the Winter Olympics, which the South will host beginning February 9.
"The government is open to talking with North Korea, regardless of time, location and form," said South Korean Unification Minister Cho Myoung-gyon.
He suggested the talks be held at the Panmunjom border village, where the 1953 armistice halting the Korean War was signed, and that they should focus on the North's participation in the athletic events. But he acknowledged other issues, including the nuclear issue, could arise.
President Donald Trump, in a Tuesday morning tweet, reacted to the possible inter-Korean talks by saying, "Rocket man now wants to talk to South Korea for first time. Perhaps that is good news, perhaps not - we will see!"
Trump also said, "Sanctions and 'other' pressures are beginning to have a big impact on North Korea."
Later in the day, Haley told reporters at the United Nations that "North Korea can talk with anyone they want, but the U.S. is not going to recognize it or acknowledge it until they agree to ban the nuclear weapons that they have."
Kim's overture might be an attempt to drive a wedge between Seoul and Washington, but State Department officials said that would not happen.
The department's spokeswoman, Heather Nauert, told reporters the United States was skeptical about Kim's sincerity in expressing a willingness for his negotiators to sit down with those from the rival South, and that it was highly unlikely that Washington would play any role in such talks.
Sanders on Tuesday declared that "our alliance and friendship with South Korea remains stronger than it ever has been, and we're in close contact with those people about a unified response" to the overture from Pyongyang.
WATCH: North Korea, South Korea talks
"We've been very clear about the fact of what our priority is and that's a denuclearized peninsula, and there's nothing new to update on that front," the press secretary said.
Last year, North Korea tested three intercontinental ballistic missiles and conducted its sixth and most powerful nuclear test.
In a New Year's Day speech, Kim said he had a "nuclear button" on his desk and that the United States should be aware that its entire territory was now "within the range of our nuclear strike."
Greta Van Susteren contributed to this report.