Over the weekend, the rebels tried to reconnect an encircled area in the east with insurgent territory in the west.
They set off a huge tunnel bomb underneath army positions in the strategic Ramouseh district.
The army has been fighting back with the help of Russian air strikes to stop the rebels breaking through.
Around a quarter of a million civilians are living under siege in rebel-held areas since government forces cut them off last month.
"We are now overlooking the Ramousah area but Russian jets are intensifying their bombing, which is holding us back from moving quickly," a rebel commander told Reuters news agency.
Aleppo was once Syria's commercial capital and also boasted a rich architectural and archaeological heritage.
Much of it has been destroyed or looted during more than five years of war.
Russia and Syria have announced the opening of what they called humanitarian corridors for civilians and rebels wanting to surrender but few people are reported to have used them.