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.
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.
"What is stopping the agreement presently is Iranian obstinacy. But the deal still stands, the ceasefire stands until now," said a commander with the rebel 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 5am, while opposition officials had been expecting a first group of wounded people to leave earlier.
