Syrian government forces have been making rapid advances into eastern Aleppo, forcing thousands of civilians to flee.
The Syrian army captured Jabal Badro on Sunday, a day after seizing Hanano district, and is making in-roads in other neighbourhoods.
Its aim is to divide the rebel-held east of the city in two.
Some reports said several thousand civilians were trying to escape, to government and other rebel-held areas.
Some 275,000 people are believed to remain in eastern Aleppo, where food and medical supplies have all but run out.
Seven-year-old Bana Alabed, who has gathered thousands of Twitter followers with her tweets from Aleppo, said on Sunday that her home in the east of the city had been bombed.
She tweeted hours earlier: "Last message - under heavy bombardments now, can't be alive anymore. When we die, keep talking for 200,000 still inside. BYE.- Fatemah"
It is hard to get a clear picture of how events are unfolding in eastern Aleppo, but it is being reported that:
- Government forces have pushed into Sakhour neighbourhood, coming within just 500m of creating a corridor through the rebel-held territory
- Reuters news agency says government troops have taken the district of Holok, while the Syrian Observatory of Human Rights says they have also taken Baadeen
- Meanwhile, Kurdish-led forces have reportedly taken the Bustan al-Basha neighbourhood, allowing some 2,500 civilians to flee north
The Syrian army said it had helped 1,500 people to flee to government-controlled western parts, while Russian news agencies said some 900 civilians - including 119 children - had left Jabal Badro alone before the troops moved in.
One man left Hanano on Saturday, telling Reuters news agency it was "because of the bombardment from the Syrian army during the advance, and the chlorine gas".
He was waiting with his wife, mother and three children at a minibus stop, hoping to travel on to the west of the city.
Analysis: By Sebastian Usher, BBC's Arab Affairs Editor
This is the biggest blow so far to rebels in Aleppo. They seem to be losing ground quickly to pro-government forces attacking them from the east.
The loss of Hanano was a strategic victory for President Assad's forces - but also a symbolic one as it was the first district rebels seized in Aleppo in 2012.
The government's aim seems to be to cut the rebel-held area in two at its narrowest point. There is heavy fighting in the Sakhour district. If the rebels lose it, the army will have achieved this objective.
The fighting has set off a wave of civilians seeking refuge not only in other rebel areas, but also on the government side.
The Syrian army's offensive to retake eastern Aleppo is now into its 13th day.
Around 219 civilians, including 27 children, have been killed in the assault so far, the UK-based Syrian Observatory for Human Rights says.
Retaking all of Aleppo would be a major victory for Syrian President Bashar al-Assad after five years of conflict.
Once Syria's commercial and industrial hub, Aleppo has been divided roughly in two since 2012, with the government controlling the west and rebels the east.
In the past year, Syrian troops have broken the deadlock with the help of Iranian-backed militias and Russian air strikes.
Russia says its air force is active in other parts of the country, but not operating over Aleppo.