The offensive, launched yesterday, aims to break through a three-month encirclement of the battered city's eastern districts, where more than 250,000 people live without access to food or humanitarian aid.
"In just a few days, we will open the way for our besieged brothers," rebel commander Abu Mustafa told AFP from the frontline district of Dahiyet al-Assad, on the southwestern outskirts of Aleppo.
Fighting and air strikes pounded nearly all of Aleppo's western outskirts, with the most intense clashes reported in the districts of Al-Zahraa and Dahiyet al-Assad.
The Syrian Observatory for Human Rights monitoring group said rebels and allied jihadists have unleashed a barrage of rocket fire and at least 10 car bombs since their assault began.
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The British-based group said two days of fighting have killed at least 20 regime forces and allied fighters as well as 26 Syrian rebels, but it did not give a toll for foreign militants battling alongside the opposition.
At least 21 civilians, including two children, have been killed in rebel bombardment since Friday morning.
The offensive has seen more than 1,500 rebels from the provinces of Aleppo and nearby Idlib amass along a front stretching for 15 kilometres down the city's western edges.
Their aim is to work their way east through a sprawling military complex, then to the district of Al-Hamdaniyeh to break through government lines.
Fighting on Saturday was so fierce around Al-Zahraa and Dahiyet al-Assad that the explosions and gunfire could be heard across Aleppo's eastern half, AFP's correspondent there said.
He said pro-government forces launched a counter-attack on Saturday and managed to recapture several positions in Dahiyet al-Assad, where rebels had scored a major advance.