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"We will see if Pyongyang's message today, that it is willing to hold talks, represents the first steps along the path to denuclearization," the statement read. "In the meantime, the United States and the world must continue to make clear that North Korea's nuclear and missile programs are a dead end."
South Korean President Moon Jae-in said in a statement on Sunday that he'd met with the North Korean delegation in Pyeongchang before the closing ceremony of the Winter Olympic Games, and told them that North Korea-US talks should happen "as soon as possible."
Moon said the North Koreans indicated they were willing to talk with the US, and agreed that "the inter-Korean relationship and North Korea-US relationship should develop together."
Moon met for an hour with the North Korean delegation at an undisclosed location in Pyeongchang at 5 p.m. local time Sunday, three hours before South Korea prepared to host the carefully-choreographed closing ceremony.
In his statement, Moon said he met all eight members of the delegation before speaking with North Korean delegation leader Kim Yong Chol, vice chairman of the Party Central Committee and Ri Son Kwon, chairman of the Committee for the Peaceful Reunification of the Fatherland.
As the lights dimmed in the stadium for the closing ceremony, delegates from North Korea and the US took their seats in the VIP box.
President Donald Trump's daughter, Ivanka Trump, who is attending the Olympics as a US presidential adviser, sat just feet away from North Korean delegation leader Kim Yong Chol.
A similar photo opportunity presented itself two weeks ago when US Vice President Mike Pence took his assigned seat for the opening ceremony, not far from Kim Yo Jong, the younger sister of North Korean leader Kim Jong Un.
At that appearance, Pence looked stony-faced when the North and South Korean athletes entered the stadium. This time, Ivanka Trump stood up to applaud the combined team.
"So the maximum pressure campaign is going to continue and intensify. But if you want to talk, we'll talk," Pence said.
The measures target the "deceptive shipping practices that have enabled the Kim regime to fund its dangerous weapons programs," Treasury Secretary Steven Mnuchin said in a statement.
Asked whether he would rule out the United States boarding and inspecting North Korean ships, Mnuchin said, "No, I cannot rule that out."
North Korea said in a statement Sunday that it considers "any type of blockade an act of war against us."
"If tension on the Korean peninsula escalates into a brink of war due to the US reckless actions, all the catastrophic consequences resulting therefrom will be borne by the United States," said the statement which was attributed to a Foreign Ministry spokesman.
US President Donald Trump said in a press briefing Friday that if the sanctions didn't work, the country would have to go to "phase two," although he didn't specify what that would be.
"Phase two may be a very rough thing, may be very, very unfortunate for the world. But hopefully the sanctions will work," Trump said.
North Korea responded with its own threats Sunday, saying that "if the US has indeed the guts to confront us in a 'rough' manner, we will not necessarily take the trouble to stop it."
As she arrived with the US delegation in South Korea Saturday, White House Press Secretary Sarah Sanders had said no talks were scheduled.
However speculation of potential North Korea-US talks rose Sunday when South Korea released the full list of North Korean delegates, which includes Choi Kang Il, the deputy director of North American affairs.
CNN's Stella Ko and Maegan Vazquez contributed to this report.