Ukraine say the theatre was hit by a Russian airstrike on Wednesday while people sheltered there from bombardments. Russia denied striking the theatre.
Italy's culture minister said his country will rebuild Mariupol's theatre, which was badly damaged earlier this week.
Writing on Twitter, Dario Franceschini said that a proposal he made to the cabinet was approved, to offer Ukraine the "resources and means to rebuild it as soon as possible."
"Theatres of all countries belong to the (of) whole humanity" he added.
Rescue workers dug survivors on Thursday out of the rubble of the theatre in the besieged city which Ukraine said had been hit by a Russian airstrike while people sheltered there from bombardments.
Russia denied striking the theatre.
However, its forces have blasted cities and killed many civilians in its assault on Ukraine, now entering its fourth week.
Mariupol has suffered the worst humanitarian catastrophe of the war, with hundreds of thousands of civilians trapped in basements with no food, water or power as Russian forces pound it with artillery fire and airstrikes.
A city mayoral adviser, Petro Andrushchenko, said the number of victims of the purported strike on the theatre on Wednesday was not known, but the shelter had held.
"Now the rubble is being cleared," he told Reuters by phone. "There are survivors."
Commercial satellite pictures showed the word "children" had been marked out on the ground in front of the building before it was hit.
Russian foreign ministry spokeswoman Maria Zakharova said the allegation that Russia had bombed the theatre was a "lie" and repeated Kremlin denials that Russian forces have targeted civilians.
"Russia's armed forces don't bomb towns and cities," she told a briefing.
Mariupol city council later said more than 350,000 people were still sheltering in the city and 30,000 had left.