It comes amid mounting tensions between the two states after the secretive communist state carried out several intercontinental ballistic missiles (ICBM) tests, prompting fears that parts of the US could now be within range of a nuclear strike.
In response, Mr Trump warned North Korea it would face “fire and fury the like of which the world has never seen” if the tests continued.
Mr Kim’s government then announced it was putting together a plan to fire four missiles towards the US Pacific territory of Guam.
While the war of words has eased somewhat in recent days, Mr Wood told a United Nations conference on disarmament that the US is considering all options for dealing with the “growing threat” posed by North Korea.
"North Korea's ballistic missile and nuclear weapons programmes pose grave threats to the entire world," he told the meeting in Geneva. "Its recent ICBM tests are another example of the dangerous, reckless behaviour of the North that is destabilising the region and beyond.
"My President's top priority remains protecting the homeland, US territories and our allies against North Korean aggression. We remain prepared to use the full range of capabilities at our disposal against the growing threat from North Korea."
North Korean representative Ju Yong Chol was present at the time but did not immediately respond.
The Pentagon has previously confirmed it is developing military options as a possible response to North Korea.
Last month, Marine General Joseph Dunford, the Chairman of the US Joint Chiefs of Staff, said it was “not unimaginable” that there would be a war between the two countries.
"Many people have talked about military options with words like 'unimaginable,'" he said. "I would probably shift that slightly and say it would be horrific, and it would be a loss of life unlike any we have experienced in our lifetimes, and I mean anyone who's been alive since the Second World War has never seen the loss of life that could occur if there's a conflict on the Korean Peninsula.
"But as I've told my counterparts, both friend and foe, it is not unimaginable to have military options to respond to North Korean nuclear capability. What's unimaginable to me is allowing a capability that would allow a nuclear weapon to land in Denver, Colorado. That's unimaginable to me. So my job will be to develop military options to make sure that doesn't happen."
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