The intensity of the conflict in Syria's biggest city and elsewhere suggests that Assad remains determined to cling to power, with support from Iran and Russia, despite setbacks such as this week's defection of his newly installed prime minister.
"We have retreated, get out of here," a lone rebel fighter yelled at Reuters journalists as they arrived in Aleppo's Salaheddine district. Nearby checkpoints that had been manned by rebel fighters for the last week had disappeared.
Syrian state television said government forces had pushed into Salaheddine, killing most of the rebels there, and had entered other parts of the city in a fresh offensive.
It said dozens of "terrorists" were killed in the central district of Bab al-Hadeed, close to Aleppo's ancient citadel, and Bab al-Nayrab in the southeast.
The military offensive appeared to be the most significant ground attack in Aleppo since rebels seized an arc of the city stretching from the southeast to the northwest three weeks ago.
Joma Abu Ahmed, an activist with the rebel Free Syrian Army, told Reuters that insurgents had fallen back to the nearby neighbourhood of Saif al-Dawla, which was now under fire from army tanks inside Salaheddine and from combat jets.
Some rebels denied retreating and an opposition watchdog, the Syrian Observatory for Human Rights, said fighting in the area was the most violent since insurgents first moved in.
"Fierce clashes are continuing inside Salaheddine district between rebel brigade fighters and the regime forces, which have stormed the district," the British-based Observatory said.
Abu Firas, a member of the Free Syrian Army, said rebels had left only one building in Salaheddine. "We did not withdraw, our guys are still there and the situation is in our favour."
The rebel Tawheed Brigade said its fighters had repelled Assad's forces trying to storm the shattered neighbourhood.
"Yesterday they were able to destroy five tanks and a MiG plane near Aleppo International Airport," the brigade's field commander Abdulkader Saleh said in an emailed statement.
As Assad's forces battle for Aleppo, there has been no let-up in fighting elsewhere in Syria. More than 240 people were killed across the country on Tuesday, 40 of them in the central city of Homs, the Syrian Observatory for Human Rights said.
Aleppo, at the heart of Syria's failing economy, has taken a fearful pounding since the 17-month-old uprising finally took hold in a city that had stayed mostly aloof from the revolt.
Satellite images released by Amnesty International, obtained from July 23 to Aug 1, showed more than 600 craters, probably from artillery shelling, dotting Aleppo and its environs.
"Amnesty is concerned that the deployment of heavy weaponry in residential areas in and around Aleppo will lead to further human rights abuses and grave breaches of international law," the human rights group said, adding that both sides might be held criminally accountable for failing to protect civilians.
The military's assaults in Aleppo follow its successful drive to retake neighbourhoods seized by rebels in Damascus after a July 18 bomb attack that killed four of Assad's closest aides, including his feared brother-in-law Assef Shawkat.