“The cabinet ... has approved my proposal to offer Ukraine the resources and means to rebuild it as soon as possible. Theatres of all countries belong to the whole (of) humanity,” Franceschini wrote on Twitter.
Ukraine said the theatre was hit by a Russian air strike on Wednesday while women and children sheltered there from bombardments.
Russia denied striking the theatre.
But 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 for weeks.
Russian forces have begun letting some people out in private cars this week but have blocked aid convoys from reaching the city.
Viacheslav Chaus, the governor of Chernihiv, a northern city that has been intensely bombarded, said 53 civilians had been killed there in the past 24 hours.
In the capital of Kyiv, a building in the Darnytsky district was extensively damaged by what the authorities said was debris from a missile shot down early in the morning.
As residents cleared glass and carried bags of possessions away, a man knelt weeping by the body of a woman which lay close to a doorway, covered in a bloody sheet.
Although both sides have pointed to limited progress in peace talks this week, President Vladimir Putin, who ordered Russia’s invasion on February 24, showed little sign of relenting.
In a vituperative televised speech, he inveighed against “traitors and scum” at home who helped the West, and said the Russian people would spit them out like gnats.
Dmitry Medvedev, the deputy head of Putin’s security council, said the United States had stoked “disgusting” Russophobia in an attempt to force Russia to its knees.
“It will not work; Russia has the might to put all of our brash enemies in their place,” he said.
Kyiv and its Western allies believe Russia launched the unprovoked war to subjugate a neighbour that Putin calls an artificial state carved out of Russia.
Moscow says it is carrying out a “special operation” to disarm it and “denazify”.
Heavily outnumbered Ukrainian forces have prevented Moscow from capturing any of Ukraine’s biggest cities so far despite the largest assault on a European state since World War II.
Ukrainian President Volodymyr Zelenskiy invoked the Berlin Wall and the Holocaust in a speech by video link to Germany’s Bundestag, a day after a similar speech to the US Congress.
Russia has assaulted Ukraine from four directions, sending two massive columns towards Kyiv from the northwest and northeast, pushing in from the east near the second biggest city Kharkiv, and spreading in the south from Crimea.
But British military intelligence said in an update on Thursday that the invasion had “largely stalled on all fronts”, and Russian forces were suffering heavy losses from a staunch and well-coordinated Ukrainian resistance.
Northeastern and northwestern suburbs of Kyiv have been reduced to rubble by heavy fighting, but the capital itself has held firm, under a curfew and subjected to deadly nightly rocket attacks.
Amid the unrelenting fighting, both sides have spoken of progress at talks.
Ukrainian officials have said they think Russia is running out of troops to keep fighting and could soon come to terms with its failure to topple the Ukrainian government.
Moscow has said it is close to agreeing a formula that would keep Ukraine neutral, long one of its demands.
- With AAP/ANSA
