The attack on multiple fronts was accompanied by heavy rebel shelling of the western side of the city, killing several civilians and destroying homes.
Once Syria's economic powerhouse, Aleppo has been ravaged by the war that began with anti-government protests in March 2011 and has killed more than 280,000 people.
The city has been divided between government control in the west and rebel control in the east since mid-2012, with the regime long seeking to sever the key Castello Road supply route into the opposition side.
The rebel offensive began at dawn, an AFP correspondent said, reporting heavy clashes between regime and opposition forces along the front line of the northern city.
The Britain-based Syrian Observatory for Human Rights also reported the assault.
"Opposition factions launched a wide-scale attack on four fronts against the regime inside Aleppo city, including in the Old City," said Observatory chief Rami Abdel Rahman.
"There is currently fierce fighting underway, but the opposition has not advanced because of the heavy aerial bombardment the regime is carrying out on the areas where fighting is underway."
A barrage of 300 rebel shells had been fired into western Aleppo since dawn, killing nine civilians, said the Observatory.
State television said eight civilians had been killed and 80 people wounded in fire by "terrorist groups on residential neighbourhoods in Aleppo city".
There was no immediate toll for either side taking part in the clashes.
Ahmed, a resident of western Aleppo, said the shelling had completely destroyed his home in the Syriac quarter.
"The shells have rained down on the western neighbourhoods since 4:30 am," he told AFP.
Residents of the neighbourhood gathered in one of the affected areas on Monday morning, lifting debris and helping locals gather their belongings so they could leave in search of shelter elsewhere.
"The sounds of the fighting can still be heard and are very loud, and the shelling on the western neighbourhoods is ongoing," he said.
Meagre food stocks
Mahmud Abu Malak, a spokesman for the Nureddin al-Zanki rebel group, described fierce fighting.
"All kinds of heavy artillery and machineguns are being used in the assault, which is intended to ease the pressure on the Mallah and Handarat fronts," he said, referring to areas near the rebel supply route into Aleppo.
The Castello Road route was effectively severed on Thursday when government forces seized a hilltop within firing range.
The advance leaves the opposition-held east of the city cut off, and raises the prospect of total siege.
On Sunday, rebels launched an assault to push government forces back from the road and reopen the route, but it failed and ended with at least 29 opposition fighters dead.
The ongoing fighting comes despite a nationwide truce declared by the government last Wednesday, to coincide with the Muslim holiday of Eid al-Fitr.
The ceasefire has produced little respite in violence, but was extended for another 72 hours on Saturday.
The severing of the Castello Road has already created shortages of food and fuel in the east of Aleppo, with local market stalls sparsely stocked.
"There are very few vegetables today because the Castello Road is closed," said Abu Mohamed, a vendor in the Bustan al-Qasr neighbourhood.
"This eggplant and zucchini is grown here inside Aleppo," he said, gesturing to his meagre stock.
"If we hadn't planted eggplant and zucchini inside the city, we wouldn't have had any vegetables at all."
Residents also described searching in vain for fuel, whether for vehicles or home use.
The situation has raised fears of a lengthy siege of the east of the city.
The UN says nearly 600,000 Syrians live in besieged areas, most surrounded by government forces, although rebels also use the method.