"The advance is going according to plan and is sometimes faster than expected," a Syrian military source told Reuters.
The Syrian army and its allies had recaptured 32 of east Aleppo's 40 neighborhoods, about 85 percent of the area, he said.
Reuters journalists, rebels and a monitor confirmed the military thrust. There were no reports the Syrian army had made significant gains.
Russia's Foreign Minister Sergei Lavrov said the Syrian army had suspended military activity to let civilians leave rebel-held areas, RIA news agency reported.
The army and its allies tried to advance on two fronts, a Turkish-based official with the Jabha Shamiya rebel group said.
"Helicopters, warplanes and rocket bombardment like every day. Nothing has changed," the official said. Despite the bombardment, "the guys are steadfast," the official added.
During a tour of Old Aleppo on December 9, which the Syrian army took control of this week, Reuters journalists counted the sound of nine air strikes in about half an hour. Fighting could be heard from other areas nearby.
The Russian air force and Iran-backed Shi'ite militias are fighting in Aleppo on the government side. Rebel leaders have given no sign they are about to withdraw as the civilian population is squeezed into an ever-decreasing area.
Russian Defense Ministry official Sergei Rudskoi said on December 11 up to 10,500 Syrian citizens had fled parts of east Aleppo still controlled by rebels in the last 24 hours. This could not be independently verified.
Syrian government and allied forces have in the last two weeks driven rebels from most of their territory in what was once Syria's most populous city. The rebels have controlled the eastern section since 2012, and Assad said in an interview published on Thursday that retaking Aleppo would change the course of the civil war across the whole country.
The Syrian government now appears closer to victory than at any point in the five years since protests against Assad evolved into an armed rebellion. The war has killed more than 300,000 people and made more than half of Syrians homeless.
Outside of Aleppo, the Syrian army declared a ceasefire in several areas around Damascus and the northwestern province of Idlib beginning on December 9 without saying how long it would last. There was no immediate comment from rebels.