The Syrian Observatory for Human Rights said rebels fired hundreds of rockets and shells into at least seven government-held areas late on Thursday.
By Friday morning, they had made small advances, the monitoring group added.
But the Syrian state news agency Sana said the assault had been repelled and more than 100 "terrorists" killed.
Aleppo, once Syria's commercial and industrial hub, has been divided roughly in half between government-held areas in the west and rebel-controlled eastern quarters for almost three years.
Fighting on the ground and government air strikes have left thousands dead, and destroyed more than 60% of the Old City, a Unesco World Heritage site.
But in recent months, the rebels have driven government forces out of several areas in the countryside to the north, as well as almost all of the neighbouring province of Idlib.
More than 230,000 people are believed to have been killed in Syria since the uprising against President Bashar al-Assad began in March 2011. Some 11.5 million others - more than half of the country's population - have fled their homes.
Rebels have made numerous attempts to seize key installations held by the government, but with little effect.
On Thursday, 13 Islamist fighting groups and al-Nusra Front, al-Qaeda's affiliate in Syria, came together to launch a co-ordinated assault on several fronts.
A statement said the aim of the new coalition, called Ansar al-Sharia (Partisans of Islamic Law), was "the liberation of Aleppo and its countryside", after which they would work with other groups to govern the city according to Islamic law.
Western-backed groups also said they were taking part in the offensive, organised through a joint operations room called Fatah Halab (Conquest of Aleppo), the New York Times reported.
The rebels launched simultaneous attacks on western districts of the city controlled by government forces, firing hundreds of rockets and shells, the Syrian Observatory reported.
The offensive unleashed on government-held western Aleppo by Ansar al-Sharia seems to be the most serious since the battle for Aleppo began.
An early test of its prospects will be one of the first apparent objectives - the loathed and feared Air Force Intelligence headquarters in Zahra, on the western edge of the city. Rebels have tried many times to capture it, and failed.
The offensive has been building up for some time, following the capture of Idlib by a newly-coalesced rebel alliance, Jaish al-Fatah (Victory Army), in March.
What's making the difference is that rebel groups, and their outside backers like Turkey, Qatar and Saudi Arabia, which were competing in the past, now seem to be pulling together - bad news for the regime.
The success of the new campaign may depend on whether Damascus is seriously determined to hang on to Aleppo. If it is, rebel progress will be slow and hard.
The UK-based group said the fighting continued into the early hours of Friday and was focused on the frontline in Zahra, a heavily-defended area that houses several major security compounds.
At least nine people were killed and dozens wounded by the rebel barrage, it added.
The military responded with a heavy aerial and artillery bombardment of rebel positions in the city, and the fighting continued for several hours.
On Friday morning, military sources told Sana and Reuters news agency that the assault had been repelled and that heavy casualties had been inflicted on the rebels.
The Syrian Observatory said the rebels had managed to seize some buildings in the north-western outskirts of Aleppo, but that the gains were not of strategic importance.
The BBC's Jim Muir in Beirut says that if the rebels do manage to capture all Aleppo, it would be a huge blow to the government.
There have been persistent reports in recent weeks that it is preparing to give up both the northern city, and Deraa in the south, the better to defend the core areas of western Syria with the limited manpower at its disposal, he adds.
But state media reported that the prime minister has just been in Aleppo, visiting front-line units and dispensing large amounts of cash to the city authorities for reconstruction and administrative expenses.
The Syrian military has carried out a series of air strikes after rebel forces launched a major assault to take control of the northern city of Aleppo.