Stephen O’Brien urged the warring parties to allow the evacuation of hundreds of people who need urgent medical care.
Rebel-held front-line areas were subject to dozens of Russian air strikes overnight, observers said.
It allowed government troops to gain ground in the north of the city.
Hours later, the Syrian army offered rebel fighters safe passage if they evacuated eastern Aleppo.
“The army high command calls all armed fighters in the eastern neighbourhood of Aleppo to leave these neighbourhoods and let civilian residents live their normal lives,” said a statement, carried by state news agency Sana yesterday.
“The Russian and Syrian military leaderships will guarantee safe passage for the fighters and will give them aid as necessary.”
The government made a similar offer in July, but the rebels ignored it.
Syrian government forces have been trying for months to recapture the city’s eastern half, which has long been a major opposition stronghold.
“I am deeply alarmed by the ferocious pummelling of eastern Aleppo city,” Mr O’Brien, UN under-secretary for humanitarian affairs, said in a statement.
“Indiscriminate bombing and shelling continues in a shocking and unrelenting manner, killing and maiming civilians, subjecting them to a level of savagery that no human should have to endure.”
The healthcare system in eastern Aleppo had been “all but obliterated”, he added.
Yesterday air strikes hit Aleppo’s main trauma M10 hospital for the third time in a matter of days, medical workers say.
“The hospital is now out of service completely,” radiologist Mohammad Abu Rajab was quoted by Reuters as saying.
The Syrian Observatory for Human Rights, a UK-based opposition monitoring group, said dozens of Russian air strikes overnight had targeted front-line “fighting zones”.
A correspondent with AFP news agency said the districts of Bustan al-Basha, Sakhur and Suleiman al-Halabi were all hit.
Observatory director Rami Abdel Rahman said the strikes had helped pro-government troops to advance in the north of the city, reaching the outskirts of al-Halaq district.
“Regime forces aims to control Bustan al-Basha and Sakhur districts, to tighten the zones controlled by rebels,” he said.
Ismael Abdullah, a volunteer with Syrian civil defence group the White Helmets, told the BBC: “The bombing started from the morning. There are air strikes in the al-Maja neighbourhood and rescue teams are there to get the people out from under the rubble.”
Syrian government forces in Aleppo’s Suleiman al-Halabi neighbourhood on September 30, 2016Image
Russia and the US are at loggerheads over ending the conflict in Syria, both supporting opposing sides in the five-year civil war.
A US-Russia brokered cessation of hostilities recently collapsed and the US had threatened to end co-operation with the Kremlin unless Russia halts its military campaign.
For its part, Russia accuses the US of secretly supporting the powerful jihadist group Jabhat Fateh al-Sham (formerly known as al-Nusra Front) in its attempts to unseat President Bashar al-Assad.
It has emerged that Russian Foreign Minister Sergei Lavrov and his US counterpart John Kerry talked again on Saturday about the situation in Aleppo.
But there was still no sign of any diplomatic breakthrough that might end the bloodshed.
Once Syria’s commercial and industrial hub, Aleppo has been divided roughly in two since 2012.
At least 250,000 people have been killed in the conflict, with the observatory estimating the true number to be about 430,000.
More than 4.8 million people have fled abroad, and an estimated 6.5 million others have been displaced within the country, the UN says.