Turkey-led rebels have advanced to the outskirts of Afrin city since launching an assault on the wider Kurdish enclave of the same name on January 20.
"More than 2,000 civilians have arrived in the area of Nubul" controlled by pro-regime forces after fleeing the enclave, said the Syrian Observatory for Human Rights.
"Several hundred more are still on the roads" heading out of the region, the Britain-based Observatory said.
Turkish forces and their allies on Saturday arrived within less than two kilometers of the city, sparking fears it could become besieged, AFP reported.
Afrin is home to around 350,000 people, the Observatory says.
The Observatory says pro-Ankara forces have captured around 60 percent of the Afrin enclave since January 20.
The YPG led a U.S.-backed alliance that succeeded in expelling Islamic State (IS) from much of Syria.
Hundreds of Kurdish and Arab forces from the anti-IS alliance have left the fight against the militant group to head to Afrin to help.
The Observatory says more than 200 civilians have been killed since the Afrin battle began.
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