The assault began on Sunday and is intended to ease the encirclement of the opposition-held east of Aleppo city, where an estimated 250,000 residents have been under regime siege since July 17.
The Britain-based Syrian Observatory for Human Rights described it as the largest rebel attack in Aleppo since 2012, when fighting reached the city and left it roughly divided between the opposition control in the east and regime forces in the west.
But government troops backed by Russian warplanes have put up a fierce defence, the monitor said.
"The Russian raids didn't stop all night on the front lines" there, said Observatory head Rami Abdel Rahman.
"This has slowed the offensive and allowed regime troops to retake five of the eight positions that rebels had taken since Sunday," he added.
The strikes came despite an appeal by US Secretary of State John Kerry on Monday for Russia to "restrain" itself and its ally in Damascus from "offensive operations".
Kerry said regime attacks had prevented the warring parties from meeting for negotiations on Monday, the target date set for the regime and opposition forces to agree on the framework of a political transition.
The Aleppo offensive groups fighters from Fateh al-Sham Front, formerly Al-Qaeda affiliate Al-Nusra Front, as well as the powerful Islamist Ahrar al-Sham and other factions.
The Observatory said 50 rebels and allied jihadists had been killed since the operation began on Sunday, as well as dozens of regime troops.
Six people also died and 10 were wounded by mortar rounds that hit regime-controlled districts of southwest Aleppo, it said.
The official SANA agency said five people died in the bombardment.
'Last chance for rebels'
The route used by regime forces and civilians living in government-controlled parts of Aleppo runs through Ramussa, on the southwest outskirts of Aleppo.
It has become the main target of the rebel assault. Its capture would both cut off government forces and open a new route into the city for rebels.
The Observatory said at least 30 civilians had also been killed since Sunday in opposition bombardment of government-held southwestern districts of Aleppo.
"This battle is the last chance for rebels. If they lose, it will be difficult for them to launch a new assault to break the siege," Abdel Rahman said.
"For the regime also, it's a question of life or death. They've been preparing for this battle for months and it'll be a tough blow for its troops if they lose."
Elsewhere in Aleppo province, the Observatory said at least 11 people, including five children, were killed in air strikes believed to have been carried out by Russian warplanes on the rebel-held town of Atareb.
The town has been targeted multiple times in recent weeks. According to the Observatory, 76 people have been killed in strikes there since July 16.
Residents in east Aleppo have reported food shortages and rising prices since government troops seized the last remaining road into opposition-held districts on July 17.
Breathing difficulties
In the town of Saraqeb, 50 kilometres (30 miles) south of Aleppo, 24 people suffered breathing difficulties after a barrel bomb attack, the Observatory said.
Residents said the attack had used chlorine gas, but the monitor could not confirm this.
The incident took place close to where Russia said on Monday one of its military helicopter was shot down over Idlib province, killing the five people on board.
Last week Russia announced the opening of "humanitarian corridors" to allow residents and surrendering fighters to flee the east for government-held territory.
The announcement was met with scepticism by both residents and some in the international community, and 35 NGOs in a statement on Tuesday called the initiative "deeply flawed".
The groups, including Save the Children and Oxfam, urged implementation of a UN call for a weekly 48-hour humanitarian pause in the city.
But Damascus and Moscow say some residents and fighters have begun using the passages and Syrian state media reported Tuesday that "dozens of families" had crossed from the east.
It reported similar crossings over the weekend, although residents and rebels in east Aleppo dismissed those reports as "lies".
Syria's conflict has killed more than 280,000 people and drawn in world powers on both sides since it erupted in March 2011.