A large majority of Americans consider North Korea's nuclear weapons program a critical threat toward the United States, according to a new poll.

However, they remain divided on which policy would best contain that threat — and, for the first time in almost 30 years, a majority of Americans were found to support military action if North Korea attacked South Korea.

The poll, conducted by the Chicago Council on Global Affairs, offers a glimpse of how Americans are responding to the rapidly evolving tensions with Pyongyang. Just two years ago, 55 percent of Americans listed North Korea as a critical threat facing the United States. Now 75 percent do, making it among the greatest perceived threats in the poll.

And notably, while many analysts now suggest that convincing North Korea to give up its nuclear weapons may be impossible, few Americans think that should be an option. Only 21 percent say they would support an agreement that saw North Korea halt its nuclear program but not give up its existing weapons (17 percent of Republicans and 26 percent of Democrats).

Even fewer — 11 percent — say they'd be willing to accept a deal that would allow North Korea more nuclear weapons.

Despite the questionable success rate of sanctions already in place, 76 percent of Americans favor increasing sanctions on North Korea with strong bipartisan agreement (84 percent of Republicans and 76 percent of Democrats), while 68 percent support placing sanctions on Chinese banks and other businesses that do business with North Korea.

North Korea denounces the latest United Nations sanctions imposed on the state. (Reuters)

There are significant differences between Republicans and Democrats when it comes to the use of force against North Korea. Notably, 54 percent of Republicans favor airstrikes against North Korean weapons facilities vs. 33 percent of Democrats. However, both the use of airstrikes and U.S. troops to attack such facilities were supported by a minority of Americans; 40 percent and 28 percent, respectively.

Majority of Americans favor supporting South Korea with U.S. troops

The percentage of Americans who favor the use of U.S. troops if North Korea invaded South Korea has increased in recent years.