Luhansk region's governor, Serhiy Gaidai, confirmed two deaths, saying 60 people were feared dead under the rubble of the school in Bilohorivka.
About 90 people had been sheltering in the building and 30 were rescued, seven of them wounded, he added.
Gaidai said a Russian plane had dropped the bomb on Saturday. His accusation could not be verified independently and there was no immediate response from Russia.
Luhansk has seen fierce combat as Russian troops and separatist fighters seek to surround government forces, just over two months since the start of the Russian invasion.
Ukraine and the West have accused Russian forces of targeting civilians and war crimes, charges Moscow rejects.
Russia's 10-week-old war on Ukraine has killed thousands, destroyed cities and driven 5 million Ukrainians to flee abroad.
Ukrainian fighters in the plant have vowed not to surrender and Russian forces are seeking to declare a victory in the prolonged battle for the vast plant in time for Monday's Victory Day celebrations in Moscow, which commemorate the Soviet Union's triumph over Nazi Germany in World War Two.
The Soviet-era Azovstal steel mill, the last holdout for Ukrainian forces in the key port city, has become a symbol of resistance to the Russian effort to capture swaths of eastern and southern Ukraine.
“Probably all 60 people who are still under the rubble of the building are dead,” he posted on his Telegram account.
Emergency services were able to rescue 30 people, seven of whom were injured, the governor added.
