The assault on rebel-held districts of the divided city has raised international concern, with UN envoy Staffan de Mistura warning east Aleppo could be "totally destroyed" by year's end.
Fierce fighting rocked several districts of the city, which has been divided between government control in the west and rebel control in the east since 2012.
The Syrian Observatory for Human Rights said government forces captured a hilltop in the Sheikh Saeed district in the south of the city, but the rebels retook other parts of the neighbourhood previously captured by the regime.
Once Syria's economic powerhouse, Aleppo has been ravaged by the war, which has killed more than 300,000 people since it began with anti-government protests in March 2011.
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More than 250,000 people remain in the east of the city, which has been under near-continuous siege since mid-July and has been pounded relentlessly since the government launched its offensive on September 22.
The rebels have responded by intensifying their rocket fire on west Aleppo, where four people were killed in the Midan neighbourhood today, state television reported.
The army said earlier this week that it would rein in its bombardment of the east, and the Observatory said there had been a reduction of the bombing that has killed hundreds and destroyed the largest hospital in the rebel-held sector.
But it has pressed on with its ground offensive, with President Bashar al-Assad saying his forces had "no option" but to expel rebels from the city unless they agreed a deal with the government.
In an interview with Danish broadcaster TV2 aired yesterday, Assad said Aleppo's "best option" would be a reconciliation deal like those the government has negotiated with rebels elsewhere.
Assad said that without an agreement, he would "continue the fight with the rebels till they leave Aleppo... There's no other option."
State television reported today that people were leaving east Aleppo through humanitarian corridors but gave no further details and showed no images.