The army said it could not verify whether the target was hit. "We will not allow violation of our aerial sovereignty," the military said in a statement.
A commander in the regional alliance backing Syrian President Bashar Assad said the drone was engaged in operations in southern Syria when it was targeted by an Israeli anti-aircraft missile.
Earlier, residents of the Upper Galilee in Israel's north reported hearing an explosion and seeing a smoke trail in the sky.
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The incident comes after the Syrian army and allied forces pounded rebel-held areas in the strategic region bordering Jordan and the Golan Heights. Thousands of civilians have fled to safer opposition areas along the border with Jordan and Israel, aid workers and rebels said.
The bombings were supported by Russian aircraft, marking the first time in a while that Russian aircraft attacked targets so vigorously in southern Syria. Aa a result, it is likely that the Israeli side of the border will likely remain particularly alert.
The Syrian army's offensive threatens to wreck a de-escalation zone agreed by the United States and Russia last year and risks drawing Washington deeper into the war. Israel and Russia agreed in the fall of 2015 on a mechanism to prevent friction in the skies over Syria. A short time later, Russia sent two squadrons of combat planes to assist the Assad regime against the rebels.
Reuters contributed to this report.