Rebel groups in the area, including Fastaqim and Nusra Front, have united under one banner called the "Army of Aleppo" to fight regime forces in east Aleppo.
Syrian rebels say they are determined to fight against a massive regime advancement in eastern Aleppo.
The advance has triggered a massive humanitarian crisis as a battle between rebels and Bashar Assad's forces intensifies and hundreds of thousands of civilians flee the area they once called home.
About 50,000 people have been displaced in parts of east Aleppo.
Zakaria Malahifji, head of the political office of the Aleppo-based Fastaqim rebel group, said no rebel group in the city will withdraw.
"This is the decision of the factions. I spoke to them about everything that was tabled and they said they would not withdraw, and other things may also happen," Malahifji said from Turkey.
He confirmed that all the groups in the area, including Fastaqim and Nusra Front, have united under one banner called the "Army of Aleppo."
The regime and its allies, Russia and Iran, said they had taken the Sheikh Saeed district in the south of the city on Wednesday, but rebels denied this.
The Syrian Observatory for Human Rights, a British-based monitoring group, said the rebels retained a third of Sheikh Saeed.
The Observatory reported that the regime forces were detaining and questioning hundreds of civilians fleeing rebel-held areas. But a Syrian military source denied that there were any arrests.
Concerned over the situation, the leaders of Russia and Turkey, two of the most powerful supporters of the opposing sides in the war, had a telephone conversation in which they discussed the need of a ceasefire, as well as the provision of humanitarian aid.
But Russian UN Ambassador Vitaly Churkin told the Security Council, "We share the grave concerns of the plight of civilians in east Aleppo, but easing their suffering won't happen by ceasing the counter-terrorist operation."