The somewhat dubious Sunday claim by the state-run media in Pyongyang, true or not, quickly racheted up tensions in both the US and Japan.
"All components of the H-bomb were homemade and all the processes. were put on the Juche basis, thus enabling the country to produce powerful nuclear weapons as many as it wants", KCNA quoted Kim as saying.
In hydrogen bombs, fusion - the merging of atoms - is used to unleash energy, whereas atomic bombs use nuclear fission, or the splitting of atoms.
But Japanese Prime Minister Shinzo Abe told reporters that he had spoken with Trump by phone and said close cooperation between their countries and South Korea was needed in face of the "escalating" situation with North Korea.
North Korea has "successfully" detonated a hydrogen bomb Sunday in what could prove to be an inflection point amid rising tensions in the Korean peninsula.
"The scale of the energy was five to six times more powerful than the fifth nuclear test", Lee Mi-Sun, head of the Korea Meteorological Administration's quake and volcano centre, told a televised briefing.
North Korea conducted two nuclear tests previous year and has since maintained a torrid pace in weapons tests, including flight-testing developmental intercontinental ballistic missiles and flying a powerful midrange missile over Japan. North Korea's last nuclear test was in September 2016.
Defying U.N. sanctions and global pressure, North Korea has been test-firing missiles at a rapid pace this year. This was also echoed by China's Earthquake Administration.
The claim came after two quakes were detected in North Korea around noon on Sunday, touching off speculation that Pyongyang has conducted its sixth nuclear test despite mounting worldwide pressure. That was revised up from an initial report of 5.6.
South Korean military has raised alert level following the reports of North Korea's apparent nuke test.
There are concerns North Korea has conducted yet another nuclear test, despite repeated global warnings not to, after a magnitude 6.3 explosion was detected in the country.
The United States Geological Survey detected an "explosion" with a magnitude of 6.3.
South Korea's main spy agency has previously asserted that it does not think Pyongyang now has the ability to develop miniaturized nuclear weapons that can be mounted on ballistic missiles. It said the test was conducted according to the plans of the North's ruling Workers' Party to establish the nation as a nuclear power. Its military said it has strengthened its monitoring and readiness while considering a variety of possible responses in collaboration with the US.
The Korea Meteorological Administration initially put the scale of the natural disaster at magnitude 5.7. and said that the seismic wave was recorded at 12.36 p.m.
Japan's meteorological agency officer Toshiyuki Matsumori displays a chart showing seismic activity detected in North Korea, during a press conference at the agency's headquarters in Tokyo, on September 3, 2017.
North Korean leader Kim Jong Un inspects missiles.
South Korea's Yonhap News agency says a magnitude 5.6 quake has occurred in North Korea.