This would require a submarine being able to sail within missile range of America.
Obama gave warning of the possible consequences. "We could, obviously, destroy North Korea with our arsenals," he told CBS News. "But aside from the humanitarian costs of that, they are right next door to our vital ally, [South] Korea."
He said that America was improving its own missile defences.
"One of the things that we have been doing is spending a lot more time positioning our missile defence systems, so that even as we try to resolve the underlying problem of nuclear development inside of North Korea, we're also setting up a shield that can at least block the relatively low-level threats that they're posing right now," he said.
Kim, who inherited the leadership of North Korea from his father in 2011, had shown himself to be "irresponsible" and "erratic", said Obama.
North Korea is believed to possess only one Sinpo-class submarine capable of launching a missile.
It would be no match for the nuclear-powered attack submarines of the US Navy.
More threatening for America is North Korea's continued development of the Taepodong-2 intercontinental ballistic missile. Once perfected, this weapon could be used to conduct a nuclear strike on the US mainland from North Korean soil.
America has responded to the build-up of Kim's nuclear arsenal by opening formal talks with South Korea - the country most directly threatened - on the deployment of advanced missile defences.
The Terminal High Altitude Area Defence System (THAAD) would offer the surest protection against its neighbour, which has run four nuclear tests.
But any deployment of the missile shield in South Korea would also change the military balance in East Asia. China adamantly opposes any introduction of missile defences in the region, fearing this would reduce the potency of its own nuclear deterrent.
America is using the possible arrival of THAAD as a way of placing pressure on China to curb North Korea's ambitions. Kim's regime is almost completely dependent on Chinese aid.
China has already sent signals of exasperation with North Korea, criticising the nuclear tests and allowing the passage of United Nations Resolution 2270, which imposed tougher sanctions on Mr Kim's regime last month.