The report by Fox highlighted a far less significant threat than those posed than the long-range missiles, known as ICBMS, that have been central to escalating tensions in the region.
In a message to the public, the governor of Guam Eddie Baza Calvo said there was now "no threat" to the island and the Marianas archipelago, but that Guam was "prepared for any eventuality".
He added that Guam is "not just a military installation", but American soil with American citizens.
A USA official says American intelligence agencies have assessed that North Korea has developed a nuclear warhead that could be fitted onto an intercontinental ballistic missile.
The president's extraordinary comments had already pushed USA stocks into negative territory on Tuesday afternoon.
Hecker says the real threat is "stumbling into an inadvertent nuclear war on the Korean Peninsula by misunderstanding or miscalculation". Inflammatory rhetoric on both sides will make that more likely.
USA crude CLc1 shed 12 cents to $49.05 a barrel, while Brent crude LCOc1 fell 18 cents a barrel at $51.96 a barrel.
While it's extremely unlikely that North Korea's leadership would risk its own destruction with a pre-emptive attack on Guam, an American territory about 3,400 kilometers (2,100 miles) from the Korean Peninsula, analysts believe the North does have several missiles in its arsenal capable of reaching the Pacific island, which Pyongyang has regularly threatened because of its USA military bases.
A top Chinese expert on North Korea said Pyongyang seemed to have been heartened by Washington's failure to take firm measures in response to earlier actions.
North Korea has made no secret of plans to develop a nuclear-tipped missile able to strike the United States and has ignored worldwide calls to halt its nuclear and missile programs.
Security and defence officials on Guam say there is no imminent threat to people there or in the Northern Mariana Islands after North Korea said it was examining its operational plans for attack.
"Guam remains safe, and I am confident in the ability of USA defences to protect our island and allies in the region", Bordallo said in a statement.
North Korean Foreign Minister Ri Yong Ho said Pyongyang "will, under no circumstances, put the nukes and ballistic rockets on the negotiating table", and would "teach the USA a severe lesson" if it used military force against North Korea.
North Korea is technically closer to Alaska, but Hawaiians are increasingly concerned about a possible attack from North Korea. The views expressed in this commentary are his own.
South Korea's benchmark Kospi index fell as much as 1%, and the Nikkei in Tokyo sank as much as 1.7%.
"If the USA does not retract its attempt to crush us to death and behave prudently, we will be ready and not hesitate to take ultimate measures", Pyongyang said in a statement published by the state-run KCNA news agency. The U.S. territory is home to Andersen Air Force Base. They hold the USA partly responsible, saying Trump's heated rhetoric is fueling the flames. They will be met with fire and fury like the world has never seen.
Since North Korea has conducted multiple tests of both nuclear weapons and ballistic missiles, the basic components for a nuclear-tipped ICBM were already known to exist.
President Trump's fiery warning to North Korea has unsettled Asian markets.
However veteran US Senator John McCain was sceptical about Mr Trump's statement, saying he was "not sure that President Trump is ready to act".
Trump issued the warning during a briefing on opioid addiction at his golf course in Bedminster, New Jersey.
Despite the advance, North Korea still must overcome technical hurdles before it will be seen to have perfected the technology.
Tensions have risen since North Korea carried out two nuclear bomb tests a year ago and two ICBM tests last month.
The official says the Defense Intelligence Agency assessment, first reported by The Washington Post on August 8, says North Korea has made a miniaturized nuclear warhead, but that it would still have to hurdle other technical issues before it could successfully deliver such a weapon.
That would mean North Korea has passed a crucial threshold on the path to becoming a full-fledged nuclear power.
The Washington Post, which was first to publish details, reported that it was the analysis of the Defense Intelligence Agency.
Officials at the agency have not commented. The Office of the Director of National Intelligence could not immediately be reached for comment.