Russian officials have said during high-level talks in Geneva that they have no intention of invading Ukraine, a top US diplomat says.
Reports say that around 100,000 Russian troops have been deployed near the border with Ukraine, prompting fears of an incursion.
Russia warned the US not to "underestimate the risks" involved in Moscow's confrontation with the West.
The US has said there would be sanctions if Russia were to attack.
EU foreign policy chief Josep Borrell Fontelles said he believed that a Russian invasion was still possible. "There are 100,000 Russian troops on the other side of the border," he said on Monday. "I suppose they haven't gone there to drink coffee!"
In a call with reporters following Monday's meeting, US Deputy Secretary of State Wendy Sherman described the talks as "frank and forthright" discussions over eight hours aimed at better understanding each side's security concerns.
Ms Sherman said that the Russian delegation had also denied plans to invade Ukraine, saying troop movements were "manoeuvres and exercise".
"But I would note that none of this was notified to anyone. It is typical that we notify each other's exercises to each other where we and they can prove that they in fact have no intention by de-escalating and returning troops to barracks," she added.
This was the first chance Russian and American diplomats had to discuss face-to-face the standoff over Ukraine and Russia's demands for Nato to step back from eastern Europe. And while little agreement appears to have been reached, both sides aired their concerns and set out their demands, with at least the possibility of talks continuing in the future.
Yet the gap between both sides remains large. The US urged Russia to de-escalate the situation and remove its troops from Ukraine's border but it received no assurance that would happen.
Russia demanded that Nato should give a cast-iron guarantee it would never offer membership to Ukraine. The US rejected this outright. The US offered some ideas for both countries to limit military exercises and missile deployments but there was no sense this would be enough for Russia.
Optimists will point to the fact the talks were business-like, they did not break up in acrimony and Russia insisted it had no intention of invading Ukraine. Pessimists will note that even after such assurances, US officials said they were still not sure if Russia was serious about finding a diplomatic solution to this crisis.
Russian Deputy Foreign Minister Sergei Ryabkov said the Russians had told their US counterparts "that all measures for the combat training of troops and forces are carried out within our territory" and that there was "no reason to fear any escalation scenario in this regard".
Ms Sherman said the US had raised several "preliminary" ideas aimed at de-escalating tensions, including setting reciprocal limits on the size and scope of military exercises and negotiations about missile deployments.
However, Ms Sherman said that the US was pushing back on Russian proposals that were "non-starters" for the US government, including Russia's demand that Nato commit to never include Ukraine in the alliance.
"We will not allow anyone to slam closed Nato's open-door policy, which has always been central to the Nato alliance," she said.
According to Ms Sherman, the US delegation told the Russians that any invasion would be met with "significant cost and consequences well beyond what they faced in 2014" when Russia annexed the peninsula of Crimea from Ukraine.
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These measures could include sanctions against key financial institutions, export controls, "enhancements of Nato force posture on allied territory" and increased security assistance to Ukraine, she added.
Mr Ryabkov said the talks had been "business-like and professional" but warned the US not to "underestimate the risks" to do with the confrontation between Moscow and the West.
The Geneva talks are the first of several meetings between US, allied and Russian officials this week, which will also include a meeting at Nato headquarters in Brussels and at the permanent council of the Organization for Security and Co-operation in Europe, which includes Russia.
Monday's meeting, however, took place without the participation of US European allies, including Ukraine, prompting reassurances from Ms Sherman and other US officials that Ukraine, Europe and Nato would be included in any decisions.
Mr Borrell said on Monday he'd been told that nothing would be agreed without the EU's "strong co-operation, co-ordination and participation".
