Opposition forces withdrew from all districts on the east side of the Aleppo river on Monday afternoon after losing Sheikh Saeed in the south of their pocket in overnight fighting, the Syrian Observatory for Human Rights said.
While Aleppo's fall would deal a stunning blow to opposition trying to remove Assad from power, he would still be far from restoring control across Syria. Swathes of the country remain in opposition hands, and on Sunday Daesh retook Palmyra.
Tens of thousands of civilians remain in opposition-held areas, hemmed in by ever-changing front lines, pounded by air strikes and shelling, and without basic supplies, according to the Observatory, a British-based monitoring group.
In the Sheikh Saeed district, an elderly couple stood lamenting their fate.
"May every son return to his mother. I have suffered that loss. May other women not endure the same," said the woman, her arms raised to the sky. "I have lost my three children. Two died in battle and the third is kidnapped," she added, as an army officer attempted to calm her.
Opposition groups in Aleppo received a US-Russian proposal on Sunday for a withdrawal of fighters and civilians from the city's opposition areas, but Moscow said no agreement had been reached yet in talks in Geneva to end the crisis peacefully.
The opposition official blamed Russia for the lack of progress in talks, saying it had no incentive to compromise while Assad was gaining ground. "The Russians are being evasive. They are looking at the military situation. Now they are advancing," he said.
The US National Security Council also said, in a message passed on by the American mission in Geneva, that Moscow had rejected a ceasefire. "We proposed an immediate cessation of hostilities to allow for safe departures and the Russians so far have refused," it said in a statement.
The Syrian army is backed by Russian war planes and Lebanese and Iraqi Shi'ite militias supported by Iran. Its advances on Monday were aided by a militia of Palestinian refugees in Syria, the Liwa al-Quds or Jerusalem Brigade, the general said.
The mostly Sunni opposition include groups backed by the United States, Turkey and Gulf monarchies as well as hardline jihadists who are not supported by the West.
A correspondent for Syria's official SANA news agency said the army had taken control of Sheikh Saeed, and more than 3,500 people had left at dawn.
A Syrian official told Reuters: "We managed to take full control of the Sheikh Saeed district. This area is very important because it facilitates access to al-Amariya and allows us to secure a greater part of the Aleppo-Ramousah road." The road is the main entry point to the city from the south.
Riad Hijab, Syria's chief opposition coordinator, said the opposition' defeat in Aleppo would not weaken the resolve of Assad's opponents, or push them to water down the demand that he quit.
"If Assad and his allies think that a military advance in certain quarters of Aleppo will signify that we make concessions, then (I say) that will not happen," he told reporters after meeting French President Francois Hollande.
The loss of Palmyra, an ancient desert city whose recapture from Daesh in March was heralded by Damascus and Moscow as vindicating Russia's entry into the war, is an embarrassing setback to Assad.
The Observatory reported that the jihadist group carried out eight executions of Syrian soldiers and allied militiamen in Palmyra on Monday while warplanes bombarded their positions around the city.
Still, Daesh made further advances around Palmyra, it and the Observatory said on Monday, including coming close to a military air base.
Another four people, including two children, were shot dead while the jihadists cleared the city, the Observatory said.
It said at least 34 people had died in air raids on an Daesh-held village north of Palmyra, and that local officials said poison gas had been used. Daesh accused Russia of the attack. Both Russia and Syria's military deny using chemical weapons.
The Russian Defence Ministry said on Monday that 728 opposition had laid down their weapons over the previous 24 hours and relocated to western Aleppo. It said 13,346 civilians left opposition-controlled districts of Aleppo over the same period.
"Displaced people are moving," said an Aleppo resident. Some were moving from areas controlled by the army to opposition areas, while others were going in the opposite direction. Some were staying at home waiting for the army.
The Observatory said that four weeks into the army offensive at least 415 civilians, including 47 children, had been killed in opposition-held parts of the city.
The Observatory said 364 opposition fighters had been killed in the eastern sector. It said opposition shelling of government-held west Aleppo had killed 130 civilians including 40 children.