One person was killed and 10 injured in shelling that started early Sunday morning and continued into the day, Donetsk city council spokesman Maxim Rovinsky told The Associated Press.
In a press conference in Kiev, Andriy Lysenko, a spokesman for Ukraine's National Security and Defense Council, said the only way for the rebels in Donetsk to save their lives would be to "lay down their arms and give up." He said the Ukrainian side hadn't seen the rebels show any real willingness to cooperate.
"If white flags come up and they lay down their arms, nobody is going to shoot at them," he said. "(But) we have not seen any practical steps yet, just a statement."
Conditions were clearly deteriorating in Donetsk, the largest rebel stronghold in eastern Ukraine. Associated Press reporters heard 25 loud explosions in as many minutes around noon. More than 10 residential buildings, as well as a hospital and a shop, were heavily damaged by shelling overnight, and several buses caught in the crossfire were still burning Sunday morning.
Rovinsky said Saturday that over 2,000 residential buildings had been damaged by shelling.
At least 300,000 of Donetsk's 1 million residents have fled as violence escalates between Ukrainian government forces and pro-Russian separatists, claiming the lives of over 1,300 people since April, according to the U.N.
"This is a real war! It's impossible to live in this city, I've been sleeping in the basement for the past week," said Inna Drobyshevskaya, a 48-year-old lawyer in Donetsk.
"We don't want Novorossiya (New Russia) for this price," she added, referring to a term used by rebels to describe the parts of eastern Ukraine seeking independence from the government in Kiev.
Rebel leader Aleksandr Zakharchenko called for a cease-fire Saturday but his request was met warily by both the Ukrainian government in Kiev and the West. Those leaders expressed concern that the move could be aimed at increasing international pressure on Ukraine to allow in a Russian aid mission. The West says that could be used as a pretext to bring Russian soldiers into Ukraine - and says 20,000 of them are massed near the Russian border with Ukraine.
Ukrainian President Petro Poroshenko and Western leaders have repeatedly accused Russia of providing arms and expertise to the rebels, something Russia denies.
In an interview with the AP on Sunday, rebel spokeswoman Elena Nikitina denounced the government as "incapable of negotiating" and said talks on the conflict could only begin if the Ukrainian army withdrew from the region.
That is unlikely, since the Ukrainian government has steadily retaken territory from the insurgents over the last few weeks, ousting them from smaller provincial cities and encircling Donetsk.
Poroshenko issued a statement late Saturday saying that Ukraine was prepared to accept humanitarian assistance in eastern Ukraine. But he said the aid must come in without military assistance, pass through border checkpoints under Ukrainian control and be an international mission.
Poroshenko said he and German Chancellor Angela Merkel discussed German participation in such a mission.
In Washington, the White House said President Barack Obama and Merkel agreed that any Russian intervention in Ukraine was unacceptable and would violate international law. Obama and British Prime Minister David Cameron, meanwhile, "expressed grave concern about reports that Russian military vehicles have crossed the border into Ukraine and that Russian armed forces are exercising for a 'humanitarian intervention,'" according to Cameron's office.
Russian-speaking eastern Ukraine was the power base for former President Viktor Yanukovych, who fled office in February after months of anti-government protests kicked off by his decision to scuttle an association agreement with the EU in favor of closer ties with Moscow. Fighting in the east began in April, one month after Russia annexed Ukraine's Black Sea region of Crimea.
Vitnija Saldava in Kiev, Ukraine contributed to this report.