Governor Serhiy Gaidai said Russian forces dropped a bomb on Saturday afternoon on the school where about 90 people were sheltering, causing a fire that engulfed the building.
"The fire was extinguished after nearly four hours, then the rubble was cleared, and, unfortunately, the bodies of two people were found," Gaidai wrote on the Telegram messaging app.
"Thirty people were evacuated from the rubble, seven of whom were injured. Sixty people were likely to have died under the rubble of buildings."
Reuters could not immediately verify the report.
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.
The besieged port city of Mariupol, a strategic target for Russian forces, has been virtually destroyed, with the last 300 civilians trapped in the bombed-out Azovstal steelworks evacuated on Saturday. Evacuation efforts will now focus on getting the wounded and medics out of the steelworks.
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.
US President Joe Biden and other G7 leaders are to hold a video call with Ukrainian President Volodymyr Zelensky on Sunday in a show of unity ahead of Victory Day, in which President Vladimir Putin typically inspects a massive military parade in Moscow.
"Putin's brutal attack is not only causing untold devastation in Ukraine, it is also threatening peace and security across Europe," British Prime Minister Boris Johnson said in a statement on Saturday pledging more military aid.
Britain pledged to provide a further 1.3 billion pounds ($US1.6 billion), double its previous spending commitments and what it said was the country's highest rate of spending on a conflict since the wars in Iraq and Afghanistan.