Medics in the city are struggling to cope with the huge number of casualties caused by the most sustained and intense aerial bombardment in years.
Supplies of medicine and blood are running low, as a three-week siege by the army begins to have an impact.
An air strike on a pumping station has also left many areas without water.
"The planes are not leaving the skies at all," Brita Hagi Hassan, president of the rebel city council, told Reuters news agency. "Life in the city is paralysed."
"Everyone is cooped up in their homes, sitting in the basements. These missiles are even targeting the basements and shelters that we'd set up to protect people."
Aleppo, once Syria's largest city and the country's commercial and industrial hub, has been divided roughly in two since 2012, with President Bashar al-Assad's forces controlling the west and rebel factions the east.
In the past year, troops have gradually broken the deadlock with the help of Iranian-backed militias and Russian air strikes. Earlier this month, they severed the rebels' last route into the east and placed its 250,000 residents under siege.
A short-lived truce brokered by the US and Russia provided them with a brief respite, but its collapse last Monday led to the Syrian military launching an offensive to take full control of the city.
Since then, the Syrian Observatory for Human Rights has documented the deaths of 248 people in rebel-held parts of Aleppo and its surrounding countryside.
Dozens of air strikes overnight killed 12 people, including three children, the UK-based monitoring group reported on Monday.
Bebars Mishal, a rescue worker from the Syria Civil Defence, alleged that aircraft were "using all kinds of weapons - phosphorus and napalm and cluster bombs".
The US, UK and France, which back the rebels, have also accused Syrian government and Russian forces of using bunker-busting bombs to destroy underground shelters, dropping incendiary weapons indiscriminately on civilian areas, and targeting war pumping stations.
Aref al-Aref, an intensive care medial worker, said hospitals in rebel-held Aleppo were "overwhelmed with wounded people" and that "things are starting to run out".
"We are unable to bring anything in... not equipment and not even medical staff. Some medical staff are in the countryside, unable to come in because of the siege," he told Reuters.
Abd Arrahman Alomar, a paediatrician who works for the Syrian American Medical Society, warned there were only 30 doctors still inside eastern Aleppo and that there was only enough fuel to run hospital generators for 20 days.
The charity Save the Children meanwhile reported that one of the main hospitals still operating in the area was directly hit with a barrel bomb on Monday.
Rescue efforts have also been hampered by the reported destruction of three Syria Civil Defence fire engines and two ambulances in air strikes in the past week.
Several charity kitchens were also closed on Monday because of the violence, while an AFP news agency correspondent said food prices had risen significantly.
White House spokesman Josh Earnest told reporters that President Obama was "deeply concerned" by the "sickening" bloodshed in Aleppo and elsewhere in Syria.
"What we have seen from the Assad regime and the Russians is a concerted campaign to strike civilian targets, to bomb civilians into submission," he said.
But a Syrian military source insisted it had no intention of letting up, telling AFP: "The air force will bomb any terrorist movements, this is an irreversible decision."
US President Barack Obama has expressed "deep concern" about the situation in rebel-held parts of Aleppo, amid an assault by Syrian government forces.