Statements from both countries come ahead of the much anticipated meeting between Trump and Putin expected to take place on the sidelines of the G20 summit in Germany.
Russian Foreign Minister Sergei Lavrov said on Thursday Moscow was seeking to clarify a claim by his US counterpart that Washington is ready to work together to set up "no-fly zones" in Syria.
"We have asked (for information). We haven't yet received a response to the question about which no-fly zones they have in mind. Because no one was ever talking about them," Lavrov said in comments carried on Russian state television.
"But I'm sure that the orientation of those comments, despite all the questions about one or another formulation, the orientation towards cooperation between Russia and the United States, that is a step in the right direction," Lavrov said.
US Secretary of State Rex Tillerson on Wednesday had said the US is prepared to discuss with Russia joint efforts to stabilise war-torn Syria, including no-fly zones.
He also added that the US wanted to discuss with Russia the use of on-the-ground ceasefire observers and the coordinated delivery of humanitarian aid to Syrians.
"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," Tillerson said in a statement ahead of this week's G20 summit in Germany.
The statement made no mention of Syrian regime leader Bashar Al Assad's future. The United States largely blames Assad for the six years of civil war and has called on him to step down.
Tillerson also said Russia had an obligation to prevent the use of chemical weapons by Assad's regime.
Washington hit a Syrian air base with a missile strike in April after accusing the Assad regime of killing dozens of civilians in a chemical attack. Syria denied it carried out the attack.
US President Donald Trump and Russian President Vladimir Putin are expected to meet on the sidelines of the G20 summit in Hamburg this week, and Tillerson said Syria would be a topic of discussion.
As the fight against Daesh winds down, Tillerson said Russia has a "special responsibility" to ensure no faction in Syria "illegitimately re-takes or occupies areas" liberated from Daesh or other groups.
In March, Tillerson said the United States would set up "interim zones of stability" to help refugees return home in the next phase of the fight against Daesh and Al Qaeda in Syria and Iraq.