Before the latest assault began, an estimated 250,000 people remained in the eastern part of the besieged city
Rebel fighters have called for a five-day ceasefire for medical and civilian evacuations from eastern Aleppo.
They said around 500 seriously wounded civilians needed to leave immediately and should be given passage to a safe area of countryside to the north of the city.
Overnight, rebels retreated from the last parts of the Old City after the army seized the districts of Bab al Hadid and Aqyul, a spokesperson for the Syrian Observatory for Human Rights (SOHR) said.
Regime forces appear closer than ever to retaking all of the city and winning their most important victory since the civil war began in 2011.
According to human rights monitors SOHR, at least 80,000 people have already fled Aleppo's eastern enclaves since the Syrian army mounted a new offensive in mid-November.
It said the figure included residents who had sought refuge in the government-held west of the city and a Kurdish-controlled enclave between the two sectors.
But it does not include those who had fled south to remaining rebel-held territory, Observatory chief Rami Abdel Rahman said.
Before the renewed assault, an estimated 250,000 people remained in eastern Aleppo, which was surrounded by government forces in mid-July.
Aleppo was Syria's largest city and its economic hub before the war.
US Secretary of State John Kerry has said efforts would continue to restart peace talks.
"We have been trying to find a way to get to the negotiating table ... but Assad has never shown any willingness," he said at a meeting of NATO foreign ministers in Brussels.
"Russia says Assad is ready to come to the table ... and I am in favour of putting that to the test," he added.
Earlier this week, Russia and China vetoed a resolution at the UN Security Council calling for a truce in Aleppo.
The proposal - put forward by New Zealand, Egypt and Spain - called for a seven day truce in the city.
Russian foreign minister Sergei Lavrov has said Moscow will hold talks with the US aimed at a total withdrawal of rebel forces from eastern Aleppo.