Kansai Electric Power said some 34 litres (8.8 gallons) of cooling water containing radioactive substances leaked out from the reactor at its Takahama plant 380 kilometres (236 miles) west of Tokyo.
"Resumption procedures related to the incident have been suspended as we are still investigating the cause," a company spokesman said, adding that there was no impact on the environment outside the plant.
The government and utility firms have been pushing to get reactors back in operation nearly five years after a huge earthquake and tsunami caused a disastrous meltdown at the Fukushima nuclear power plant.
Last month, another reactor at Kansai Electric Power's Takahama plant was switched on, but the accident stalls plans to bring the next one online which have already met with stiff opposition from local residents.
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But many Japanese remain wary and thousands of former residents have refused to return to areas hit by the Fukushima meltdown over fears of radiation exposure.