Ukraine has said Russian shells landed close to where radioactive material is stored, but no leaks were reported.
The nuclear power facility is the largest of its kind in Europe, with six reactors. Any damage could pose a serious health and environmental hazard.
Ukrainian President Volodymyr Zelenskyy accused Russia of "unconcealed nuclear blackmail."
"Only the complete withdrawal of Russians from the territory of the Zaporizhzhia NPP and the restoration of Ukraine's full control over the situation around the plant will guarantee the restoration of nuclear safety for all of Europe," he said during his Thursday evening address.
Russian-installed local officials instead claimed Ukrainian forces had shelled the plant for the second time in one day, disrupting the shift changeover of power plant workers.
The Zaporizhzhia plant area was also hit by shelling last weekend, with each side trading blame.
The facility in Southern Ukraine on the Dnieper river has been under Russian control since the beginning of March. It is still run by its Ukrainian technicians.
Ukrainian Foreign Minister Dmytro Kuleba called for the International Atomic Energy Agency (IAEA) to send a mission to the plant to assess the safety situation as soon as possible.
IAEA chief Rafael Grossi said there is currently "no immediate threat" to the safety of the plant due to military actions, but "this could change at any moment."
UN Secretary-General Antonio Guterres demanded an immediate end to military activity near the plant, warning that it could "lead to disaster."
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