The Syrian Army and its allies are reportedly advancing in southeastern Aleppo.
An opposition spokesman says rebel fighters face the choice of “death or surrender”.
Aleppo’s besieged rebel enclave was pounded by heavy shelling and air raids from midnight on Saturday.
A reporter with the Reuters news agency says the explosions were coming at a rate of more than one a minute.
A Syrian military source says two districts southeast of Aleppo’s ancient citadel have been taken.
“This is the first time. I have been here for five years. I heard that people are returning, so I came back,” said Mohammed Lababidi, a store owner in old Aleppo.
Mohammed Shaaban, the former director of Aleppo Museum, says he is shocked, saddened and surprised.
“I came before but it was not damaged like this. They have destroyed civilisation, they have destroyed the heritage in the oldest archaeological site on earth,” he said.
Thousands of refugees are still pouring from the areas of fighting.
The UK-based Syrian Observatory for Human Rights says more than 120,000 civilians left the city as the government advance closed in.
However, tens of thousands still remain.