Up to 1,500 people were expected to leave the Barzeh district in the north-east of the capital on Monday, a monitoring group said.
Most are being transported to the rebel-held northern province of Idlib.
The deal comes two days after powers backing opposing sides in Syria's war created four "de-escalation" zones.
Russia and Iran - allies of President Bashar al-Assad - signed a memorandum with Turkey, a key opposition supporter, calling for:
