Martin Indyk, former United States ambassador to Israel and former special envoy for Israeli-Palestinian negotiations, said on Twitter Assad appeared to be offering to "get the Iranians to pull back from Golan" Heights, if Israel accepts it as Syrian. "We believe that there is no place for any Iranian military presence, anywhere in Syria".
Israel is vehemently opposed to Iran maintaining a military presence in Syria, however.
The Syrian regime's survival, largely due to Russian intervention, placed Vladimir Putin directly at odds with Israeli Prime Minister Benjamin Netanyahu.
A Syrian state-run newspaper says Damascus will keep fighting the country's rebels - or "terrorists" as the government calls opposition fighters - despite USA warnings against a new offensive in southern Daraa province.
Along with Idlib province in the north, Daraa is one of the last areas of territory still held by rebels after a Syrian government push to recapture areas lost during the seven-year war.
The US has also warned Assad not to move troops into the region.
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In addition to violence inside Syria, Daraa's location on the border of the Israeli-occupied Golan Heights makes any operation there highly sensitive.
Chagai Tzuriel, director general of the Israeli intelligence ministry, said he believed Russian Federation was growing frustrated with Iran's presence in Syria and was anxious that fighting between Israel and Iran would threaten Moscow's interests.
According to the report, Israel's Defense Ministry underlined Israel's view that the Iranian presence should be entirely eradicated from Syria, not only militarily and not only from the southern part of the country, but also from an economic standpoint as well.
A senior Israeli official made clear that Netanyahu's government would not deem the exclusion of Iranian forces from the border region sufficient.
Jordanian officials served as mediator, shuttling messages between the two rooms, according to the report.
On May 10, Israel launched a large-scale attack on what it said were Iranian-operated targets in Syria, raising fears of a major confrontation.