Washington (CNN) The Trump administration is considering military action against North Korea if the rogue regime successfully builds a nuclear missile capable of hitting the United States, according to multiple sources with knowledge of the administration's latest thinking. Senior national security officials believe a nuclear armed Pyongyang represents an unacceptable risk to the US.

Beyond the missile threat to the US homeland, the national security officials pushing for military action believe that if North Korea becomes a full nuclear power, it will proliferate, potentially sharing nuclear and missile technology with states such as Iran, Pakistan and Libya, and non-state actors.

Such a prospect would increase the nuclear threat to US interests. That possibility, according to these officials, is ultimately more dangerous to the US and the region than conventional military action to thwart Pyongyang's nuclear ambitions, despite the risk of significant civilian and military casualties in such a conflict.

"All the damage that would come from a war would be worth it in terms of long-term stability and national security," Senator Lindsey Graham, a South Carolina Republican who has had multiple conversations with President Donald Trump about North Korea, told CNN.

On Thursday Trump spoke with President Moon Jae-in of South Korea, and the White House readout of the call stated that the two leaders "noted their firm position that any dialogue with North Korea must be conducted with the explicit and unwavering goal of complete, verifiable, and irreversible denuclearization."

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