A Syrian military source said the army and its allies had captured the al-Asila and Aaajam districts southeast of Aleppo's ancient citadel.
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”.
Large numbers of civilians are said to be under heavy bombardment.
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.
Gunfire was also heard.
— Reuters Top News (@Reuters) December 11, 2016
A Syrian military source said the army and its allies had captured the al-Asila and Aaajam districts southeast of Aleppo’s ancient citadel.
The southern part of the Karam al-Daadaa neighbourhood has also reportedly been taken.
The army also seized the al-Maadi district on Sunda morning.
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 remain.
Why is Aleppo important?
The fall of the whole city to President Bashar al-Assad would bring him the biggest victory yet after nearly six years of civil war.
However, Assad would still be far from restoring government control over the entire country.
Aleppo – who is fighting who?
Assad’s forces are supported by Russian air power, Iran and Shi’ite militias from Lebanon, Iraq and Afghanistan.
Together, they have helped to turn the tide of conflict his way after he seemed to be on the back foot in mid-2015.
The mostly-Sunni rebels include groups supported by the US, Turkey and Gulf monarchies.
There are also some jihadist factions that receive no assistance from the West.
Russian and US officials met in Geneva on Sunday for more talks on an elusive deal for civilians and fighters to leave the city.
There has reportedly been no word on their progress.
ISIL fighters staged a surprise assault on Palmyra over the weekend.
The UK-based monitor the Syrian Observatory for Human Rights says the militants recaptured the ancient city on Sunday after re-entering it on Saturday.
What they are saying
“The result will certainly be a complete end of the district, in a tragic way,” – a Turkish-based official from the Jabha Shamiya rebel group in Aleppo.
“Embrace death for Aleppo!” – graffiti on a wall in the city’s ancient citadel.