The plant's off-site power lines, vital lines of defence against potential nuclear meltdown, have already been cut and the shelling at Enerhodar has caused a lasting blackout there.
That has prompted Ukraine to say it may have to shut down the last operating reactor supplying power to Zaporizhzhia including the cooling systems for the plant's nuclear fuel.
"This is an unsustainable situation and is becoming increasingly precarious. Enerhodar has gone dark. The power plant has no off-site power. And we have seen that once infrastructure is repaired, it is damaged once again," International Atomic Energy Agency chief Rafael Grossi said in a statement.
Ukraine and Russia have blamed each other for shelling near Zaporizhzhia in southern Ukraine and within the perimeter of Europe's biggest nuclear power plant, which has six reactors.
"This is completely unacceptable. It cannot stand," Grossi said.
"I therefore urgently call for the immediate cessation of all shelling in the entire area. Only this will ensure the safety and security of operating staff and allow the durable restoration of power to Enerhodar and to the power plant."
Grossi this week called for the creation of a "nuclear safety and security protection zone" around Zaporizhzhia, repeating his call on Friday.
Zaporizhzhia's operator is not confident that off-site power can be restored and that is prompting it to consider shutting down the last operating reactor, Grossi said.
"The entire power plant would then be fully reliant on emergency diesel generators for ensuring vital nuclear safety and security functions. And as a consequence, the operator would not be able to re-start the reactors unless off-site power was reliably re-established," he added.
Russia said it backed Grossi's call.
"We fully support the appeal and demand of the #IAEA Director General that shelling of the town of Enerhodar and the #ZNPP must stop immediately," its ambassador to the IAEA Mikhail Ulyanov said on Twitter.
