Implementation of the deal, brokered by Russia and Turkey, was expected to begin around 5:00 am (0300 GMT), the Syrian Observatory for Human Rights and sources on the ground said.
But three hours later, a dozen green government buses that had gathered at the edge of the Salaheddin neighbourhood to transport evacuees were still parked with their drivers asleep inside.
There was no immediate official explanation for the delay from the government or the rebels, or from the deal's co-sponsors.
The agreement was announced late Tuesday, as government forces were on the verge of recapturing all of east Aleppo, in rebel hands since 2012.
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In the last parts of the city still held by the rebels, large crowds gathered from before dawn awaiting evacuation.
Many spent the night in the street despite a storm, as they had fled from other areas and had nowhere to stay.
Tens of thousands of civilians have stayed on in the ever-shrinking rebel enclave for fear of arrest or torture by government forces.
Many more have fled to government-held districts or to territory controlled by Kurdish fighters.
Under the evacuation deal, both civilians and rebel fighters are to be transported to rebel-held territory elsewhere in northern Syria.