The governor of the Turkish border town of Kilis said 35,000 refugees had reached the border area - up from an estimated 20,000 on Friday.
Turkey says it is prepared to help the refugees but the frontier remains shut.
They are fleeing a Syrian government offensive on rebel-held positions near the northern city of Aleppo.
In the past few days, the Syrian army backed by Russian air power has made a series of gains in the area.
The Syrian Observatory for Human Rights, a UK-based monitoring group, says about 120 fighters on both sides were killed around the town of Ratyan, north of Aleppo on Friday.
Turkey is providing food, shelter and blankets to thousands of civilians who are stranded on the Syrian side of the border because of the fighting.
However the Turks have so far refused to open the border crossing.
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EU Enlargement Commissioner Johannes Hahn on Saturday urged Turkey to open the frontier.
"The Geneva convention is still valid which states that you have to take in refugees." Mr Hahn said, as EU foreign ministers prepared to discuss the crisis in Amsterdam.
On Thursday 60 donor countries meeting in London pledged billions of dollars to ease the plight of Syrian refugees - with the EU agreeing to spend $3.3bn (€3bn; £2.3bn).
About 4.6 million people have fled Syria during the five-year civil war. Another 13.5 million are said to be in need of humanitarian assistance inside the country.
Turkey hosts the largest number of refugees - 2.5 million.