In a wide-ranging statement, Secretary of State Rex Tillerson said Russia had a "special responsibility" to help create stability on the ground -- or risk hobbling the fight against the Islamic State group.
Despite their "unresolved differences on a number of issues", Tillerson made a strong case for the two nations to work together in Syria.
"The United States is prepared to explore the possibility of establishing with Russia joint mechanisms for ensuring stability, including no-fly zones, on the ground ceasefire observers, and coordinated delivery of humanitarian assistance," he said.
Tillerson issued his statement before joining Trump in Europe, where the US leader will meet Putin for the first time on Friday on the sidelines of a G20 summit in Hamburg, Germany. Syria will loom large in their discussions.
It also came as US-backed fighters inch forwards in Raqa's Old City, in what Washington sees as a "key milestone" in the campaign to defeat the Islamic State group in its de facto Syrian capital.
"ISIS has been badly wounded, and is could be on the brink of complete defeat in Syria if all parties focus on this objective," Tillerson said. "In order to complete the mission, the international community, and especially Russia, must remove obstacles to the defeat of ISIS."
"We call upon all parties, including the Syrian government and its allies, Syrian opposition forces, and Coalition forces carrying out the battle to defeat ISIS, to avoid conflict with one another and adhere to agreed geographical boundaries for military de-confliction and protocols for de-escalation."
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No progress on safe zones
Powerbrokers Russia, Turkey and Iran failed Wednesday to iron out the details of four safe zones in war-torn Syria after two days of peace talks in Kazakhstan.
Moscow and Tehran, which back President Assad, and rebel supporter Ankara agreed in May to establish four "de-escalation" zones in a potential breakthrough after more than six years of fighting that has claimed more than 320,000 lives.
While fighting dropped off in the weeks after the deal, it has since returned in several areas, and the international players have yet to agree the exact boundaries of the zones or determine who will police them.
Chief Russian negotiator Alexander Lavrentiev said seven documents outlining how the zones should work "need finalising" despite being "essentially agreed" between the three key powerbrokers in Astana.
"We have not yet managed directly to establish the de-escalation zones," Lavrentiev said, insisting however that "de-facto" safe zones already exist on the ground.
Lavrentiev said that the borders of two zones -- covering rebel-held parts of Homs province and Eastern Ghouta near Damascus -- have been agreed in principle.
One major stumbling block appears to be over who will ensure security in all four areas, with Turkey and Iran in particular reportedly wrangling to bolster their influence.
Lavrentiev said that it had been agreed that Russian military police carrying light weapons would be involved in patrolling buffer areas around some of the zones.
He also accepted that a zone meant to cover swathes of southern Syria would likely need to involve the United States and Jordan in some capacity.
A source close to the Syrian rebel delegation that attended the talks told AFP that they would refuse a proposal to have Iran monitor the safe zone in central Homs province.
Turkish and Russian forces are likely to be deployed in the northern de-escalation zone's "buffer territory, separating the opposition and regime" in parts of Idlib and neighbouring Aleppo province, the source suggested.
The lack of progress shows how tough the going has been for Moscow as it pushes the Astana talks in a bid to turn its game-changing intervention on the side of Assad into a concerted push to pacify Syria.
More meetings
A working group of experts from Russia, Iran and Turkey is set to meet in Tehran on August 1-2 to try to thrash out the details, before another round of meetings in Astana later that month.
The talks in Astana have largely seen the West sidelined, but they are intended to complement broader political negotiations the United Nations is backing in Geneva, which are due to restart next week.
Talking in Astana, UN Syria envoy Staffan de Mistura said there had been "small progress" in the work on de-escalating the conflict and called for "progress on the political side".
Russia has argued the zones agreement will provide moderate rebels with security and help focus attacks against jihadist groups such as former Al-Qaeda affiliate Fateh al-Sham Front and the Islamic State group.
Moscow on Wednesday flexed its muscles on the ground by firing a barrage of its latest X-101 cruise missiles at IS targets in Syria from strategic bombers.
Russia's defence ministry said the strikes had destroyed three jihadist arms depots and a command post on the border between Homs and Hama provinces.
