The North's official Korean Central News Agency (KCNA) said its military was "carefully examining the operational plan for making an enveloping fire at the areas around Guam with medium-to-long-range strategic ballistic rocket Hwasong-12".
It said the plan would be reported to the Supreme Command after "full examination and completion" and put into practice at the order of leader Kim Jong-un, South Korea's Yonhap news agency reported.
The tiny but important island of Guam
The 541 sq km (209 sq miles) volcanic and coral island in the Pacific between the Philippines and Hawaii.
It is an "unorganised, unincorporated" US territory, with a population of about 163,000.
US military bases cover about a quarter of the island. About 6,000 personnel are based there and there are plans to move in thousands more.
The heated rhetoric between the US and North Korea intensified after Pyongyang tested two intercontinental ballistic missiles (ICBM) in July, claiming it now had the ability to hit the mainland US.
Mr Trump told reporters on Tuesday: "North Korea best not make any more threats to the US. They will be met with fire and fury like the world has never seen."
Veteran US Senator John McCain was sceptical about Mr Trump's statement.
"The great leaders that I have seen, they don't threaten unless they are ready to act and I'm not sure that President Trump is ready to act," said Senator McCain.
Analysis: Words with consequences?
Anthony Zurcher, BBC News, Washington:
Donald Trump said that North Korean leader Kim Jong-un "has been very threatful, beyond a normal statement". So he responded with language that goes well beyond a normal statement for any US president.
Perhaps Mr Trump believes that no hyperbolic threats should go unmatched or that apocalyptic warnings are the only ones the North Korean leadership will understand. Perhaps he - intentionally or not - is pursuing a Nixonian "madman" style foreign policy, where adversaries will tread lightly to avoid triggering the wrath of an unpredictable US commander-in-chief.
When the leader of the world's greatest superpower, the only nation ever to have used nuclear weapons on an enemy, talks of unprecedented "fire and fury", however, those words have consequences.
During his presidential campaign Mr Trump criticised his predecessor Barack Obama for not enforcing a red line against Syria's use of chemical weapons. Now President Trump has drawn a fiery bright line of his own with North Korea - one that could commit the US to a perilous course of action if his words go unheeded.
North Korea reacted angrily after fresh sanctions were announced on Saturday by the UN following repeated missile tests.
The sanctions aim to reduce North Korea's export revenues by a third.
It called the sanctions a "violent violation of our sovereignty", the news agency said.
China, which is Pyongyang's closest ally, has said it is "100%" committed to enforcing the latest round of sanctions.
Russia and China have previously differed with others on how to handle Pyongyang, but in recent months have joined calls for North Korea to stop its missile tests - while also urging the US and South Korea to halt military drills, and withdraw an anti-missile system from the South.Pyongyang's news agency says a plan to hit the US territory could be enforced at "any moment" once Kim Jong Un makes a decision.
By Connor Sephton, News Reporter
North Korea says it is "carefully examining" a plan to strike the US territory of Guam with missiles.
A spokesman for the Korean People's Army said the strike plan will be "put into practice in a multi-current and consecutive way any moment" once Kim Jong Un makes a decision.
He added that "enveloping fire" would be used to contain major US military bases on the island territory in the western Pacific Ocean - including the Anderson Air Force Base.
KCNA, Pyongyang's state-run news agency, also carried a statement from a different military official which said North Korea may carry out a pre-emptive operation if the US shows signs of provocation.
Earlier on Tuesday, the Washington Post had reported that North Korea has successfully made a miniaturised nuclear warhead that can fit inside one of its intercontinental ballistic missiles.
The newspaper said that claim was contained in a confidential assessment by America's Defence Intelligence Agency.
An analyst who specialises in North Korea, who asked not to be named, told Sky News that he believes Pyongyang is "a lot more than halfway" to achieving its aims of producing a nuclear weapon capable of exploding above the US mainland.
He added: "It's fair to say it's just a matter of time, in terms of being able to hit the hit the US with a (nuclear) missile."
Pyongyang said the sanctions were caused by a "heinous US plot to isolate and stifle" the country - and its officials also threatened to make America "pay the price for its crime... thousands of times".