"The operation's target is not against a country or a person but against terrorist organisations only," Erdogan told local neighbourhood chiefs in Ankara.
"No one should have any doubts or take our statements to mean something else. Even if Turkey is left alone, it will continue its fight against terrorist organisations."
Turkey launched its unprecedented operation in northern Syria in August, providing tanks and aerial support to opposition fighters as they retake territory from the Islamic State (IS) group.
The next day, Assad's close ally Moscow which is providing key military support to Damascus, demanded an explanation.
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The Syrian foreign ministry also denounced the comments, saying they showed "clearly that the flagrant Turkish aggression against the Syrian territory is only the result of the ambitions and the illusions of an extremist despot".
Moscow and Ankara have become closer since ties were restored this summer after Turkey shot down a Russian plane on the Syrian border in November last year.
Experts say rebels supported by Ankara are a varied collective of different Syrian opposition brigades rather than a single organised force.
So far, they have been retaken IS territory including Jarabulus, Al Rai and the symbolically important town of Dabiq.
Although the offensive was targeting jihadists, it has the secondary purpose of stopping the advance of Syrian Kurdish groups: the Syrian Kurdish Democratic Union Party (PYD) and People's Protection Forces (YPG) militia.
Ankara views the YPG/PYD as sister organisations of the outlawed Kurdistan Workers' Party (PKK) waging an insurgency inside Turkey since 1984.
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