Rebel fighters tell Al Jazeera that they still cling on to territories taken from government forces recently.
Syrian rebels have rejected claims that government forces and allies, including fighters from the Lebanese Hezbollah movement, have taken back parts of Aleppo recently captured by rebels.
Hassan al-Eshra, a fighter in the Jaysh al-Islam armed group, said that it and other rebel groups were still in control of the flashpoint neighbourhood of Ramosa on Tuesday evening, despite government attempts to recapture it.
"We haven’t fallen back from any of the sites ... we took from the regime three days ago," Eshra told Al Jazeera by telephone from the frontline. "Yesterday the regime tried [to recapture Ramosa] three times, but it is strategically incapable."
Syrian rebel groups have said that they are fighting to hang on to key areas in the northern city as government forces and allies escalated attacks.
Eshra told Al Jazeera that rebels were still in control of the Telat al-Snobarat area on the southwestern edge of the city.
"This area is still under our control," he said, adding that rebels targeted an industrial area and gas field north of the city with several missiles.
Fatah Halab, a coalition of national and local rebel groups, broke the government-imposed siege on the city over the weekend along with Jaish al-Fatah, or the Army of Conquest.
The Army of Conquest is a coalition of rebel groups that includes Jabhat Fatah al-Sham (formerly the al-Nusra Front).
The Syrian government has sent thousands of reinforcements to the city in recent days as part of an attempt to reverse gains made by rebel groups.
Hezbollah’s Al Manar television channel and Al Mayadeen, a news channel close to the Syrian government, reported earlier on Tuesday that pro-government fighters, supported by heavy air strikes, had recaptured Telat al-Snobarat.
Earlier on Tuesday, the United Kingdom-based Syrian Observatory for Human Rights reported that at least nine people were killed in government strikes on eastern, rebel-controlled Aleppo.
Injuries were also reported in the government-held part of the city as rebel groups fired shells into the area.
Baraa al-Tello, a rebel fighter from the Harakat Fajer Shahba faction of the Nour Al-Din Al-Zenki armed group, recalled the offensive that broke a government siege on the city three days earlier.
"I was part of the the latest attack when we broke the siege," he told Al Jazeera. "War planes, helicopters and every kind of weapon that exists were used by the regime to hit us."
Although claiming rebels still controlled Telal al-Snobarat, Tello said: "The fighting is still very furious, but all of our men are doing their best to advance there and in other areas."
People across the city set tyres ablaze and burned other items to create a de facto no-fly zone and impair the ability of government fighter jets and helicopters to hit their targets.
The United Nations on Tuesday called for humanitarian access to the city amid fears that trapped civilians would go short of food and water.
"These cuts are coming amid a heatwave, putting children at a grave risk of waterborne diseases," Hanaa Singer, UNICEF’s representative in Syria, said in a statement.
The statement said Aleppo’s civilians, particularly children, could face a "humanitarian catastrophe" if fighting intensified.
Fayez Sandeh, a resident of Aleppo’s Bustan al-Qasr neighbourhood, said very little food or medicine had come into the city in recent days as fighting continued.
"Fuel hasn’t entered yet and the electricity isn’t regularly [there]," he told Al Jazeera. "We haven’t had water for more than 10 days. We use well water, but it’s not healthy for humans. We don’t have any other options."
The Syrian conflict started with peaceful protests against President Bashar al-Assad in March 2011, but it quickly evolved into a full-on civil war that has raged for more than five years.
The Syrian Observatory estimates that more than 280,000 people have been killed, while efforts to negotiate a resolution have repeatedly failed.