The Zaporizhzhia power plant was occupied by Russian troops in March and has been shelled in recent weeks, with Ukraine and Russia blaming each other for the attacks.
Damage to the active zone of the reactor would "have consequences not only in Ukraine but also definitely beyond its borders" said head of the Ukrainian nuclear security agency Oleg Korikov during a press conference.
Zaporizhzhia nuclear plant is using internal power supplies since being disconnected from the grid on Monday.
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The plant may have to use "diesel power plants to provide electricity to its security systems" according to Korikov, who warned that "it is very difficult to replenish diesel fuel in wartime conditions."
The main risk would then lie in "running out of diesel, and this can lead to an accident damaging the active zone of the reactor and to the release of radioactive elements in the environment" Korikov said.
On Tuesday, the UN atomic agency (IAEA) published a report on the "untenable situation" at the nuclear power plant.
Ukraine shares a land border with Moldova, Romania, Hungary, Slovakia, Poland, Belarus and Russia.