A ceasefire agreement brokered by Russia, Assad's most powerful ally, and Turkey ended years of fighting in the city and has given the Syrian leader his biggest victory yet after more than five years of war, Reuters reported.
Officials in the military alliance fighting in support of Assad could not be reached immediately for comment on why the evacuation was delayed.
Rebel sources said the ceasefire remained in place despite the delay in the evacuation plan.
Read alsoBattle of Aleppo ends after years of bloodshed with rebel withdrawal - Reuters"What is stopping the agreement presently is Iranian obstinacy. But the deal still stands, the ceasefire stands until now," said a commander with the Nour al-Din al-Zinki group, speaking in a voice message to Reuters from eastern Aleppo.
Sources on Tuesday had given different expected start times for the evacuation. A military official in the pro-Assad alliance had said the evacuation was due to start at 5 a.m. (10:00 p.m. ET), while opposition officials had been expecting a first group of wounded people to leave earlier.
Read alsoUN: Aleppo civilians 'shot on the spot'However, none had left by dawn, according to a Reuters witness waiting at the agreed point of departure. Twenty buses were waiting there with their engines running but showed no sign of moving into Aleppo's rebel-held eastern districts.
"There is certainly a delay," said Rami Abdulrahman, director of the Syrian Observatory, a war monitor.
Officials with Aleppo-based rebel groups accused Shi'ite militias backed by Iran of obstructing the Russian-brokered deal. The pro-opposition Orient TV cited its correspondent as saying the plan may be delayed until Thursday.
People in eastern Aleppo have been packing their bags and burning personal possessions as they prepare to leave, fearing looting by the Syrian army and its Iranian-backed militia allies when they restore control.