The United Nations said some 20,000 people have gathered at the Bab al-Salam crossing, hoping to reach Turkey, which already hosts more than two million refugees from the bloody conflict.
The Syrian Observatory for Human Rights for its part estimated some 40,000 people had been forced to leave their homes since last Monday.
Turkish authorities were working to free up space within the existing camps along the Syrian border to accommodate the new arrivals.
An AFP correspondent saw trucks carrying parts for tents yesterday to the refugee camp close to the border gate on the Turkish side which faces the Bab al-Salam crossing on Syrian soil.
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At least four Turkish aid trucks were also seen returning to Turkey after making deliveries of food to the Syrian side of the border.
Turkey's Disaster and Emergency Management Authority (AFAD) said in a statement that it had finalised preparations for a possible influx.
Turkey faced a similar experience in 2014 when 200,000 refugees fled the Syrian Kurdish town of Kobane over three days as the Islamic State group and Syrian Kurdish fighters battled for control.
The government offensive is targeting the Aleppo province, which was once a rebel stronghold, providing easy access to neighbouring Turkey, a key opposition backer.
The city itself has been divided between rebel control in the east and government control in the west since mid-2012.
But government forces have steadily chipped away at rebel-held territory around the city and their advances this week leave the opposition there virtually surrounded.
The advance is the most significant outcome yet of the Russian intervention that began on September 30, ostensibly targeting the Islamic State group and other "terrorists".