The statement came after Russia announced it was pulling back some troops from exercises that have raised fears of a potential invasion of Ukraine.

Speaking after talks with German Chancellor Olaf Scholz, Mr Putin said the US and NATO rejected Moscow's demand to keep Ukraine and other ex-Soviet nations out of NATO, halt weapons deployments near Russian borders and roll back alliance forces from Eastern Europe.

But the US and NATO have agreed to discuss a range of security measures that Russia had previously proposed.

Mr Putin said Russia was ready to engage in talks on limiting the deployment of intermediate-range missiles in Europe, transparency of drills and other confidence-building measures but emphasised the need for the West to heed Russia's main demands.

The statement followed the Russian Defence Ministry's announcement of a partial pullback of troops after military drills, adding to hopes the Kremlin might not invade Ukraine imminently. The Russian military gave no details on where the troops were pulling back from, or how many.

Mr Scholz agreed diplomatic options are "far from exhausted".

The announcement of troops being pulled back was a "good signal," he said, adding that he hopes that "more will follow."

The announcement buoyed world financial markets and the long-suffering ruble after weeks of escalation in Europe's worst East-West standoff in decades. It came a day after Russia's Foreign Minister indicated the country was ready to keep talking about the security grievances that led to the Ukraine crisis — a gesture that changed the tenor after weeks of tensions.

Russia says some troops going back to base

Hours before the Russian Defence Ministry statement on the troops, a US defence official said Russian units were moving closer to the Ukrainian border – not away from it.

And Western officials continued to warn that the Russian military could attack at any time, with some floating Wednesday as a possible invasion day. NATO's chief said the alliance had no proof yet of a Russian retreat.

The White House declined to offer immediate comment on Russian troop movements.

The fears of an invasion grew from the fact that Russia has massed more than 130,000 troops near Ukraine. Russia denies it has any such plans, despite placing troops on Ukraine's borders to the north, south and east and launching massive military drills nearby.

US and other NATO allies, meanwhile, have moved troops and military supplies toward Ukraine's western flank, although not to confront Russian forces, and promised more financial aid to the ex-Soviet nation.

Moscow brandished Tuesday's pullback announcement as proof that fears of war were fabricated by a hostile, US-led West.

"February 15, 2022, will go down in history as the day Western war propaganda failed. Humiliated and destroyed without a single shot fired," Russian Foreign Ministry spokesperson Maria Zakharova tweeted.

Yet Ukraine remains effectively surrounded on three sides by military forces from its much more powerful neighbour, and even if the immediate threat recedes, longer-term risk remains. Russia annexed the Black Sea peninsula of Crimea from Ukraine in 2014, and some 14,000 people have been killed in fighting between Ukrainian troops and pro-Russia separatists in the country's east.

The Russian Defense Ministry did not indicate where the withdrawing troops had been deployed or how many were leaving.

It released images of tanks and armoured vehicles rolling onto a train, and a tank commander saluting his forces while a military band played. The ministry did not disclose where or when the images were taken, or where the military vehicles were headed, other than "to places of permanent deployment."

Foreign Minister Sergey Lavrov said the troops were returning "according to plan." He said such drills always adhered to a schedule — regardless of "who thinks what and who gets hysterical about it, who is deploying real informational terrorism."

'We'll believe when we see': Ukraine

Ukraine's leaders expressed scepticism.

"Russia constantly makes various statements," Ukrainian Foreign Minister Dmytro Kuleba said.

"That's why we have the rule: We won't believe when we hear, we'll believe when we see. When we see troops pulling out, we'll believe in de-escalation."

The new glimmers of hope accompanied a flurry of diplomacy. German Chancellor Olaf Scholz planned to meet with Russian President Vladimir Putin in Moscow, a day after he sat down with Ukraine's leader in Kyiv.

Foreign Minister Zbigniew Rau of Poland, one of Russia's most strident European critics, was also in Moscow on Tuesday to meet Mr Lavrov, while Ukraine's Foreign Minister hosted his Italian counterpart.

Mr Lavrov's comments came at a made-for-TV meeting with Putin and seemed designed to send a message to the world about the Russian leader's position.

The Foreign Minister argued Moscow should hold more talks, despite the West's refusal to consider Russia's main demands.

The talks "can't go on indefinitely, but I would suggest to continue and expand them at this stage," Mr Lavrov said, noting that Washington has offered to discuss limits for missile deployments in Europe, restrictions on military drills and other confidence-building measures.

Moscow wants guarantees NATO will not allow Ukraine and other former Soviet countries to join as members. It also wants the alliance to halt weapons deployments to Ukraine and roll back its forces from Eastern Europe.

Mr Lavrov said possibilities for talks "are far from being exhausted."

Mr Putin noted that the West could try to draw Russia into "endless talks" and questioned whether there was still a chance to reach agreement. Mr Lavrov replied that his ministry would not allow the US and its allies to stonewall Russia's main requests.

"The path for diplomacy remains available if Russia chooses to engage constructively," White House principal deputy press secretary Karine Jean-Pierre said.

"However, we are clear-eyed about the prospects of that, given the steps Russia is taking on the ground in plain sight."

One possible off-ramp emerged this week: Ukraine's ambassador to the UK, Vadym Prystaiko, pointed at a possibility of Ukraine shelving its NATO bid — an objective that is written into its constitution — if it would avert war with Russia.

Mr Prystaiko later appeared to back away from the idea, but the fact that it was raised at all suggests it is being discussed behind closed doors.

Even amid the hopeful signs, the US and European countries kept up their warnings.

British Foreign Secretary Liz Truss reiterated on Tuesday that the danger of an invasion still exists, telling Sky News that it "could be imminent".

But she added that "there is still time for Vladimir Putin to step away from the brink."

Norwegian Foreign Minister Anniken Huitfeldt issued a similar warning.

US says Russia continuing preparations

US officials have said the Russian military continued apparent attack preparations along Ukraine's borders. A US defence official said small numbers of Russian ground units have been moving out of larger assembly areas for several days, taking up positions closer to the Ukrainian border at what would be departure points if Mr Putin launched an invasion.

The official spoke on condition of anonymity to discuss information not publicly released. CBS News was first to report on the movement of the units.

Maxar Technologies, a commercial satellite imagery company that has been monitoring the Russian buildup, reported increased Russian military activity in Belarus, Crimea and western Russia, including the arrival of helicopters, ground-attack aircraft and fighter-bomber jets at forward locations.

The photos taken over a 48-hour period also show ground forces leaving their garrisons and combat units moving into convoy formation.

Still, Ukrainian security and defence council chief Oleksiy Danilov downplayed the threat of invasion but warned of the risk of "internal destabilisation" by unspecified forces.

"Today we do not see that a large-scale offensive by the Russian Federation can take place either on (February) 16th or the 17th," he told reporters.

"We are aware of the risks that exist in the territory of our country. But the situation is absolutely under control."

As if to show defiance, Ukrainian President Volodymyr Zelenskyy said Wednesday would be a "day of national unity," calling on the country to display the blue-and-yellow flag and sing the national anthem in the face of "hybrid threats".

"It is not the first threat the strong Ukrainian people have faced," Mr Zelenskyy said Monday evening in a video address to the nation.

"We're calm. We're strong. We're together."

The country is preparing nonetheless. Kyiv residents received letters from the mayor urging them to "defend your city" and signs appeared in apartment buildings indicating the nearest bomb shelter.

The capital has about 4500 such sites, including underground parking garages, subway stations and basements, the mayor said.

Dr Tamara Ugrich said she stocked up on grains and canned food and prepared an emergency suitcase.

"I don't believe in war, but on TV the tension is growing every day, and it's getting harder and harder to keep calm," she said.

"The more we are told not to panic, the more nervous people become."