"He wants the world, and particularly he wants the United States, to understand that he can hit the United States and that he can hurt the United States," Feltman said. "That point is clear, even if the button thing is a gimmick and clearly not true."
Last month, Feltman became the highest-ranking U.N. official to visit the North in the past six years when he engaged in talks over several days with the country's minister and vice minister for foreign affairs.
In Kim's remarks on Monday, he also signaled that he was open to sending a delegation to the Olympic Games in Pyeongchang, South Korea, next month. South Korea responded by suggesting direct, high-level talks with North Korea on Jan. 9.
Today, military representatives from both countries held a phone call on a line that had been dormant for two years.
The U.N. General Assembly adopted a resolution in November calling for a truce during the Olympics.
"We do have a General Assembly resolution, an Olympic truce resolution, which I hope the [North Korean] leadership is looking at and thinking, the whole world is behind a truce, having a peaceful atmosphere for those Pyeongchang Olympics," Feltman said.
"We were making the point that what they see as deterrence, what they see as a strength, can actually be the risk," Feltman said, "[and] actually provoke the very war they claim they're trying to prevent."
At the heart of Kim's New Year's message is a "self-confidence," Feltman said, that he also noticed during last month's talks.
"Certainly when I was there, the interlocutors I met projected confidence that their country is acquiring the type of deterrence," Feltman said, "that allows them to negotiate from a position of strength."
"The self-confidence that I ... felt when I was there must reflect the self-confidence of the leadership of [North Korea]," he added, "allowing Kim Jong Un to signal some kind of willingness to talk to the Republic of Korea about the Pyeongchang Olympics -- and de-escalation of tensions on the Korean Peninsula."
Despite the possibility of direct talks between the North and South, the White House said Tuesday that U.S. policy "hasn't changed at all."
"The United States is committed and will still continue to put maximum pressure on North Korea to change and make sure that it denuclearizes the peninsula," White House Press Secretary Sarah Sanders said Tuesday. "Our goals are the same and we share that with South Korea. But our policy and our process has not changed in this."
Sanders declined to comment on the possibility of North Korean athletes participating in the Olympics.
ABC News' Ben Siegel contributed reporting from Washington.