Rami Abdul-Rahman, the head of the Britain-based group, said rebels belonging to an Islamist group attacked several Syrian military checkpoints and that government forces fighting back for control of the area.
The state-run news agency SANA has not commented on the fighting in the area or the clash with Israel.
A number of mortar shells have landed in the Golan over the past week, and early this month, Syrian tanks accidentally crossed into a buffer zone along the frontier of the Golan for the first time in nearly 40 years. Israel captured the Golan, a strategic plateau, from Syria in the 1967 Mideast war and subsequently annexed it.
Israel responded for the first time on Sunday, firing what it called a "warning shot" into Syria after another mortar shell strayed across the frontier and landed near an Israeli military post. Israel also warned of a tougher response if the attacks persisted.
While Israel appeared eager to calm the situation, its response was a potent reminder of how easily the Syrian civil war — already spilling across borders with Turkey, Lebanon and Jordan — could explode into a wider regional conflagration.
Israel has little love for Assad, who has provided refuge and support to Israel's bitterest enemies through the years. But the Syrian leader — and his father before him — have kept the frontier quiet for nearly four decades, providing a rare source of stability in the volatile region.
Associated Press writers Suzan Frazer in Ankara, Turkey, and Mehmet Guzel in Ceylanpinar, Turkey, contributed to this report.
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