The North's official KCNA news agency said Kim Jong-un signed off on the order at a late-night meeting of top generals.
According to the Australian Broadcasting Corporation (ABC), he called the appearance of the two B-2 Spirit bombers an "ultimatum" and told the generals:
"The time has come to settle accounts with the US imperialists in view of the prevailing situation."
Targets include US bases in Hawaii, Guam, South Korea and Japan.
Kim Jong-un "finally signed the plan on technical preparations of strategic rockets of the KPA [Korean People's Army], ordering them to be stand-by for fire so that they may strike any time," the KCNA report said.
"If they make a reckless provocation with huge strategic forces, the Korean People's Army should mercilessly strike the US mainland, their stronghold, their military bases in the operational theatres in the Pacific, including Hawaii and Guam, and those in South Korea."
According to the ABC, North Korea's arsenal includes Soviet-era Scud missiles that can hit South Korea.
Reuters wrote that its longer-range Nodong and Musudan missiles were untested, although they could in theory hit US Pacific bases.
The B-2s, meantime, were flown from the US and back in the US said was a first-time exercise designed to show America's ability to conduct long-range, precision strikes "quickly and at will."
The BBC quoted US Defence Secretary Chuck Hagel as saying Thursday:
"The North Koreans have to understand that what they're doing is very dangerous. We must make clear that these provocations by the North are taken by us very seriously and we'll respond to that."
Pyongyang, meantime, has been angered by fresh UN sanctions and recent annual US-South Korea military drills.