United States defence chief James Mattis says the United States "not looking to the total annihilation" of North Korea, but "we have many options to do so".
Mr Mattis warned Kim Jong-un to remember that the world was united against him before delivering the clearest expression yet that America was capable of destroying the North Korean regime.
Media reports say that the test was the most powerful of the six, but there is no official measurement yet of the force of the hydrogen weapon.
It has always been accepted that North Korea has intercontinental ballistic missiles (ICBMs) capable of reaching the mainland of the United States.
Trump's remarks came on Sunday after Pyongyang's announcement of a "perfect success" with its sixth and most powerful nuclear test to date.
Trump also said South Korea's "talk of appeasement with North Korea will not work".
For years, the problem with any USA military action against North Korea's nuclear program has been the mass of conventional artillery Pyongyang keeps in range of Seoul, a metropolitan area of 25 million people.
Such a move would have serious ramifications, resulting in a halt to United States trade with China, who remains its key economic partner.
"I did speak with the president", Mnuchin also said Sunday.
David Albright of the Institute for Science and International Security was skeptical that the Kim regime had successfully tested an actual hydrogen bomb but noted that their latest test yielded a more devastating blast than previous attempts.
"We are not looking to the total annihilation of a country, namely North Korea".
If North Korea hasn't already developed a hydrogen bomb then it is likely it will, before long, unless dramatically stopped.
North Korea just tested their most powerful nuclear bomb yet and caused an quake 10 times more powerful than their last test. "But as I said we have many options to do so".
China has previously said any sanctions against North Korea must be agreed through the United Nations Security Council, which was due to meet on Monday.
Trump had promised to bring "fire and fury" if North Korea continued to test its nuclear program. "It also underlines yet again the urgent need for the worldwide community to act on putting in place a legally binding ban on nuclear testing once and for all".
"The Trump Administration could announce that it is prepared to intercept future North Korean missile tests, when feasible", Bennett said. On Sunday, however, his government denounced North Korea's latest test and vowed to push for more United Nations sanctions aimed at further isolating Seoul's communist neighbor.
She expects members of congress to receive another briefing on the Korea situation as soon as this week.