During his daily call with reporters, Kremlin spokesman Dmitry Peskov declined to comment on Tillerson's remarks.
On July 4, a representative of Syria's opposition in Astana said the armed opposition disagrees with a proposal for Iran's military to monitor the areas because Iranian forces are armed combatants who fight on the side of the Syrian government's troops.
"This cooperation over de-confliction zones process is evidence that our two nations are capable of further progress", Tillerson said.
"The United States and Russian Federation certainly have unresolved differences on a number of issues", Tillerson acknowledged, but "if our two countries work together to establish stability on the ground, it will lay a foundation for progress on the settlement of Syria's political future".
"We have not yet managed directly to establish the de-escalation zones", Aleksandr Lavrentyev said after two days of meetings in the Kazakh capital. Damascus agreed to this approach but the exact borders and mechanisms for observing a truce inside those zones are still being negotiated. Syrian government forces, with the support of Russian Federation, have attacked and bombed civilians, killing and injuring thousands; maintained lengthy sieges on civilian areas; subjected tens of thousands to enforced disappearances and extrajudicial executions; and systematically tortured and otherwise ill-treated detainees, causing countless deaths in custody.
The next high-level global meeting on Syria in Astana will be held in the last week of August 2017, according to the three mediators of the peace talks (Russia, Iran, Turkey) announced in a joint statement on Wednesday.
Moscow also suggested other CIS (Commonwealth of Independent States) should consider sending peacekeeping missions to Syria.
Special units of Russian military police, equipped with light arms, will be deployed to Syria's de-escalation zones to provide safety and security there.
He said Moscow needs to make sure no faction in Syria "illegitimately re-takes or occupies areas" liberated from Daesh or other groups.
The issue is one of the few things Trump officially has on his agenda during his meeting with Putin in formal talks at the G-20 Summit in Hamburg, Germany.