The Zaporizhzhia plant in southeastern Ukraine has been occupied by Russian forces since March, and Kyiv has accused Moscow of basing hundreds of soldiers and storing arms there.
The facility has come under fire repeatedly in the past week, raising the spectre of a nuclear catastrophe.
"What is happening there is outright nuclear terrorism, and it can end unpredictably at any moment," said Dmytro Orlov, the mayor of Energodar city where the plant is based.
"The risks are increasing every day," he told AFP by telephone from the Ukrainian-controlled city of Zaporizhzhia.
He said there was mortar shelling on the plant "every day and night".
"The situation is hazardous, and what causes the most concern is that there is no de-escalation process," he added.
During his televised address on Saturday, Ukrainian President Volodymyr Zelensky accused Russia of nuclear "blackmail" and using the plant to "intimidate people in an extremely cynical way".
He added Russian troops were "hiding" behind the plant to stage bombings on the Ukrainian-controlled towns of Nikopol and Marganets.
But pro-Moscow officials in the occupied areas of Zaporizhzhia blamed the shelling on Ukrainian forces.
Missiles fell "in the areas located on the banks of the Dnipro river and in the plant", said Vladimir Rogov, a member of the Moscow-installed administration, without reporting any casualties or damage.
The river divides the areas occupied by Russia and those under Ukraine's control.
Orlov said over the past 24 hours, Energodar -- which he left at the end of April -- was shelled for the first time leading to a dramatic increase in those hoping to evacuate.
Amid safety fears, he warned that in the "near future" there may not be enough personnel to man the station.
Kyiv and Moscow have traded accusations over several rounds of shelling on the plant this month, with the strikes raising fears of a nuclear catastrophe.
In the village of Vyshchetarasivka, on the opposite bank of the Dnipro to the plant, resident Viktor Shabanin said the latest developments had left people "nervous".
"Often the wind blows in our direction. So the radiation will go immediately to us, and the radiation will go into the water," the 57-year-old added.
AFP correspondents on the ground heard air raid sirens and distant strikes on Sunday but reported no fresh fighting around the facility.
The UN Security Council held an emergency meeting over the situation on Thursday and warned of a "grave" crisis unfolding in Zaporizhzhia.
The alarm over Zaporizhzhia has revived painful memories of the 1986 Chernobyl disaster -- the world's worst nuclear accident -- that struck Ukraine when it was part of the Soviet Union and spread radioactive dust and ash across Europe.
Anastasiya Rudenko believes her late husband, who worked to decontaminate the Chernobyl disaster zone, died of bladder cancer in 2014 due to radiation.
"We could have the same fate as the people of Chernobyl," the 63-year-old told AFP.
"There's nothing good in what's going on, and we don't know how it will end."
Ukraine said the first strikes on August 5 hit a high-voltage power cable and forced one of the reactors to stop working.
Then strikes on Thursday damaged a pumping station and radiation sensors.
Backed by Western allies, Ukraine has called for a demilitarised zone around the plant and demanded the withdrawal of Russian forces.
UN grain ready to leave
Russian forces trying to press their offensive near the Dnipro in the southern Kherson region are under pressure after strategically important bridges were damaged, a Ukrainian politician said on Sunday.
Regional lawmaker Sergiy Khlan said the pontoons the Russians are using cannot fully meet their needs and that command centres were being moved as they risked being cut off from supplies.
A major consequence of the war has been soaring food prices after a Russian naval blockade and Kyiv's mining of its ports prevented Ukrainian grain from being sold on global markets.
A landmark deal last month between Russia and Ukraine brokered by Turkey and the United Nations created safe corridors to allow key grain exports to resume.
And Kyiv on Sunday said the first UN-chartered vessel transporting grain from Ukraine to relieve the global food crisis was loaded with 23,000 tonnes of wheat and is ready to depart.
Ukrainian Infrastructure Minister Oleksandr Kubrakov said the MV Brave Commander, currently in the Black Sea port of Pivdennyi, will head to Africa with a 23,000-tonne cargo of wheat.