Both residents and activists described the clashes as “unprecedented”, and said rebel fighters were attacking government positions on several fronts.
The fighting had spread to previously peaceful districts, they added.
On Thursday, the UN said the number of Syrians fleeing to other states could exceed 700,000 by the end of the year.
More than half a million are believed to have already crossed into Turkey, Jordan, Lebanon and Iraq, but only 294,000 have registered with the UN.
UN agencies and other humanitarian groups have issued a funding appeal for $488m (£300m) to help them meet the needs of the refugees.
Activists said the start of the rebel offensive in Aleppo was announced in calls from mosques at about 16:00 (13:00 GMT) on Thursday.
In interviews with foreign media and videos posted online, members of the Tawhid Brigade said a “decisive” battle for control of the city had begun. The offensive involved hundreds of rebel fighters attacking government positions on several fronts, they added.
“We wanted to surprise the Syrian army which had started to creep forward towards the southern neighbourhoods,” Bashir al-Haji, the Tawhid Brigade’s commander, told the Guardian newspaper.
“There are 6,000 fighters of the Tawhid Brigade taking part in the battle now, in addition to a few other brigades like al-Fatah and Ahfad al-Fatihin for the Turkmen.”
He denied the FSA had declared “decisive” battles for Aleppo before.
“We are not aiming to liberate the whole of Aleppo with this battle but to regain control of most of the city and get back as many neighbourhoods as we can.”
The rebel claims could not be immediately verified, but activists and residents reported heavy clashes and shelling in the districts of Izaa, Saif al-Dawla, Sulaimaniya, on Thursday night and Friday morning.
“The fighting is unprecedented and has not stopped since Thursday,” said Rami Abdul Rahman of the Syrian Observatory for Human Rights, a UK-based activist group, told AFP news agency.
“The clashes used to be limited to one or two blocks of a district, but now the fighting is on several fronts.”
Source: BBC News