"We won't take action against the Assad regime, and you get the Iranians out", the official, who requested anonymity, quoted Netanyahu as telling Putin during a meeting in Moscow.
Israel fired a Patriot missile at an unmanned drone that entered its airspace from Syria on Wednesday, the military said, warning it would not tolerate such infiltrations, which have occurred several times over the past year.
Netanyahu said Thursday that Israel would continue its formal policy of non-interference in the Syrian civil war, so long as the 1974 Separation of Forces ceasefire agreement over the Golan Heights was upheld, while continuing to act against Iran and Hezbollah.
Separately, Hossein Amir-Abdollahian, diplomat and senior foreign affairs advisor to the Iranian parliament's speaker, stressed that Iranian advisers would remain in Syria "notwithstanding aggressive Israeli attacks", Al-Watan reported.
"We are still looking into why it crossed - whether it was on a military mission and crossed on objective, or it strayed", he said, while adding that the latter scenario was "not common".
Israel did not comment on this bombing, but it was believed that their air force was targeting the T-4 Airport's barracks because of an alleged Iranian military presence.
The two leaders will also watch together the World Cup semifinal between England and Croatia scheduled for late Wednesday.
Israel is particularly anxious about the presence of Damascus ally Iran next door in Syria.
"There should be no military forces in the area of separation other than those of UNDOF", a United Nations spokesman said.
There were no immediate reports of casualties, it said.
Rami Abdel Rahman, head of the Britain-based Syrian Observatory for Human Rights, said that a missile bombardment killed an unspecified number of "Iranian and pro-regime fighters" at the T-4 base and was "likely to be Israeli".
The drone was identified and tracked by Israel's aerial defense systems and later intercepted by an Apache helicopter around 4 a.m. near the town of Beit She'an in northern Israel, the IDF said.
On July 6, Israel struck a Syrian post that it said had shelled the Golan buffer zone.