Israel purportedly rejected a previous plan that would have seen Iranian forces kept about 60 miles from its border.
While the West, Israel and Jordan welcomed the evacuation of the White Helmets, Syria and Russia view the group as an enemy.
A 1974 truce prohibits Israel's and Syria's air forces from crossing their de facto border, something Israel has frequently done.
Despite recent rifts in their strategic relationship, Russia and Iran are still working together in what appears to be the latter stages of Syria's civil war.
As the Syrian military retakes the city where a nationwide rebellion was sparked, Israel prepares to enforce the land it seized in 1967.
Syria and Israel have been technically at war since 1948 and have fought at least three major conflicts, including in 1967 when Israel seized the Golan Heights.
Thousands of Syrian refugees are camping out close to the Israeli border, under the assumption Assad won't attack so close to Israel's domain.
Report says Trump wants a deal with Russia to get the U.S. out of Syria "ASAP," but the Kremlin says it has "no idea" what CNN is talking about.
Opposition negotiator Nasr al-Hariri said such "U.S. silence" could only be due to a "malicious deal" made between foreign powers involved in Syria's war.
Putin praised the Russian military's success in Syria, saying the "army clearly demonstrated its growing potential and better coordination among military units in the struggle with terrorists."
Syria is reclaiming one of its final rebel-held regions, but Israel has reportedly continued to strike Iranian assets in the heart of the war-torn country.
The U.S.-led coalition is tasked with battling ISIS, but it has occasionally clashed with pro-Syrian government forces also battling the jihadis.
Following the reports on Russia and Israel's agreement, a top Iranian general initially brushed aside the possibility of withdrawal.
The U.S.-led coalition battling ISIS has denied responsibility for the strikes.
"We have made up our minds. There will be no retreat from the principles of the revolution or surrender of a single inch of the Syrian south."
Netanyahu accused Iran of bringing some 80,000 Shiite fighters to Syria, from countries such as Afghanistan and Pakistan, to "convert" Sunni Muslims in Syria and launch attacks against Israel.
Assad also said that Russia does not dictate his government's actions.
"He is no longer immune. His regime is no longer immune," the Israeli leader said.
Syrian President Bashar al-Assad said only "people that you trust, people who are level-headed, people who are thoughtful, people who are moral, ethical" must be listened to.
President Donald Trump called for Arab partners to contribute more to the fight against ISIS in Syria, but Saudi backing could further complicate the war.
The U.S. may reportedly dismantle its Al-Tanf base in exchange for a withdrawal of both pro-Assad and anti-Assad militias, leaving the Syrian army in control.