Russia's Foreign Minister Sergei Lavrov described his two-hour discussion with his US counterpart, Antony Blinken, as "constructive, useful" and told reporters that Washington had agreed to respond to Russia's demands over NATO positions next week, Euronews reported.
He, however, refused to characterise Washington's promise of a written response as a positive step.
“I can’t say whether we are on the right path or not. We will see when we get the American responses," he said.
Blinken, who described their meeting as a "candid exchange of concerns and ideas", stressed however that the US intends "not only to respond to the concerns that Russia has raised but to share our concerns, which are many."
He said the written response would be sent after extensive consultations with allies and added he expects the two sides "to meet again after Russia has had an opportunity to look at this paper and then we'll see where we go from there."
The meeting in Geneva, Switzerland, sought to de-escalate the crisis at the Ukrainian border and came 11 days after the first round of discussions in the Swiss city yielded little result.
Russia accuses the West of plotting “provocations” in Ukraine, citing the delivery of weapons to the country by British military transports in recent days.
Russia has amassed some 100,000 troops on the border and is being accused by the US of planning an invasion, which it denies.
Russia wants binding security guarantees, including a permanent prohibition on Ukrainian membership in NATO, to which Kiev aspires, and the removal of most of the US and allied military presence in Eastern Europe.
Just 20 minutes before the start of the meeting, Moscow announced that it wants foreign NATO troops out of Romania and Bulgaria as part of any treaty.