The warning came as North Korea said it had made a breakthrough in its pursuit of a long-range missile capable of striking the US mainland. South Korea says the North has yet to develop a functioning inter-continental ballistic missile.
Kim issued the order for the tests "in a short time," according to the Korean Central News Agency. The KCNA report did not say if Kim gave specific dates for the tests.
Some of the North's recent rhetoric was seen intended for a domestic audience to display government strength ahead of a major meeting of the ruling party in May.
In the past, North Korea has typically conducted nuclear tests and rocket launches every three to four years.
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Even if the tests happen, analysts in Seoul said the nuclear warhead explosion that Kim referred to will likely be just a test of a warhead containing only a trigger device but lacking plutonium or uranium. They said the North could also launch shorter-range missiles, but not one with an intercontinental range.
The drills, set to run until late April, are the largest ever.
Last Wednesday, North Korea's main newspaper printed photos of what appeared to be a mock-up of nuclear warhead. State media on Friday quoted Kim as having ordered more nuclear explosion tests but again didn't say when they would occur.
Kim said "a nuclear warhead explosion test and a test-fire of several kinds of ballistic rockets able to carry nuclear warheads will be conducted in a short time to further enhance the reliance of the nuclear-attack capability," KCNA reported.
Information from secretive, authoritarian North Korea is often impossible to confirm and there is virtually no way to check how genuine its claims are on developing re-entry vehicle technology.
Seoul's Defense Ministry described North Korea's claims as "unilateral," saying the country has not developed re-entry vehicle technology. Spokesman Moon Sang Gyun said the assessment is based on an analysis of South Korean and US intelligence. He declined to elaborate.
While the North might test-fire shorter-range missiles with ranges of 1,000 kilometers (620 miles) or less, it is unlikely that it would test a rocket with intercontinental range as that would probably have to fly over neighboring countries, said Jin Moo Kim, an analyst at the government-funded Korea Institute for Defense Analysis in Seoul.