Recapturing east Aleppo would be the government's biggest victory yet in Syria's five-year conflict and deal a potentially decisive blow to the opposition.
The city was once the country's economic powerhouse, but it has been ravaged by the war that has killed 300,000 people since it began with anti-government protests in March 2011.
Military aircraft dropped leaflets over east Aleppo, urging rebels to distribute food to civilians, leave the area and allow residents to do so too.
A week into the latest round of fighting for the city, the regime controls around a third of the district, the Observatory said.
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The area has been shelled heavily during the war, and many residents had already fled, but the latest fighting prompted even the last holdouts to leave.
Milad Shahabi, a member of the local council, told AFP that residents were fleeing to southern parts of the opposition-controlled east.
Observatory head Rami Abdel Rahman said that if regime forces manage to take the district they will be able to "cut off the northern parts of rebel-held Aleppo from the rest of the opposition-held districts".
At least 143 civilians, including 19 children, have been killed in the city's east since the latest assault began on November 15, according to the Britain-based monitor.
Another 16 civilians, including 10 children, have been killed in rebel fire on government-controlled western Aleppo, it said.
It added that residents of eastern districts had suffered from "suffocation" after four barrel bomb attacks and that medical officials suspected a chlorine gas attack.