The United States has said it needs "firm evidence" that Russia was withdrawing its forces from the borders of Ukraine, following President Vladimir Putin's order for his forces to pull back.

A senior US official, speaking on condition of anonymity, said that such a move would be welcome if it occurred.

But the official added that so far there was no evidence that a pull back was taking place.

Russia said that it had ordered the end of military exercises near its border with Ukraine, days ahead of a crucial presidential vote aimed at bringing the country out of crisis.

In a move that could ease tensions, Mr Putin's office said he had ordered thousands of Russian troops deployed in border regions to return to barracks after the end of spring exercises.

The US official insisted that the troops had never been conducting "exercises" near the border.

"They've been up in the border in a menacing posture," he said, and cautioned that so far "we haven't seen evidence of them" withdrawing.

"We've heard from Russian leaders in the past that they were removing troops from the border and they haven't done so," the official said, adding that Washington would closely track the situation over the next few days.

"We'll want to see clear, firm evidence of this move before we make any judgement," the official said.

Earlier, NATO Secretary General Anders Fogh Rasmussen said he had no evidence that Russian troops were withdrawing from border positions.

"Unfortunately, we have not seen any evidence at all that Russia has started withdrawal," Mr Rasmussen said, noting this was Mr Putin's third such statement on the troops, which NATO estimates number around 40,000.

On each occasion, "we have not seen any withdrawal," he said, adding that such a move would be an "important contribution to de-escalating the crisis".

The presence of Russian troops on the border has stoked serious concern after Russia annexed the Crimea peninsula in March and appeared ready to intervene to support other pro-Moscow separatist groups.

Mr Rasmussen told a briefing that Russian actions in Ukraine had brought about "a completely new security situation in Europe" after 20 years in which NATO had seen no "imminent threat".

"Now we have seen that Russia reserves the right to intervene ... it is not just words ... (and) we have to adapt accordingly," he said.

In response to this "more dangerous" state of affairs, he urged NATO member states to halt cutbacks and instead live up to commitments to spend more on defence.

Mr Putin ordered military forces to return to their permanent bases this morning after drills in three regions bordering Ukraine, the Kremlin said.

Mr Putin's office said he had issued the order because the spring manoeuvres were over.

The move could also be intended to ease tension in Russia's standoff with the West over Ukraine before it holds a presidential election on Sunday.

The Kremlin said in a statement that Mr Putin had ordered his defence chief to return troops that had been involved in exercises in the border provinces of Rostov, Bryansk and Belgorod to their "places of permanent deployment".

It was not immediately clear how many soldiers would be moved away from border regions as a result of Mr Putin's order.

Mr Putin also said on 7 May that forces had been withdrawn from the frontier, but NATO and the US said there were no signs of reduction.

In another conciliatory signal, Mr Putin welcomed what the Kremlin said were initial contacts between the Ukrainian government and "supporters of federalisation" who want more power for largely Russian-speaking regions in eastern Ukraine.

Russia wants Kiev to speak directly to the separatists.

Mr Putin also reiterated Russia's demand that Kiev end what the Kremlin calls a "punitive operation" against the separatists and pull back its troops, suggesting the pro-Western government is to blame for the violence.

He is due to meet UN Secretary-General Ban Ki-moon in China tomorrow to discuss the Ukraine crisis and other issues.

Ukrainian soldier killed during rebel attack

A Ukrainian soldier was killed when rebels staked out in a kindergarten shelled a military checkpoint near the rebel flashpoint of Slaviansk today, the defence ministry said.

Three other troops were also injured in the pre-dawn attack, which came almost a week after seven soldiers were killed in an ambush between Slaviansk and Kramatorsk.

It was the heaviest single loss of life on the part of the military in weeks of fighting in the east of Ukraine.

"The terrorists, acting in their usual cynical and insidious manner, launched the attack from a kindergarten near the railway station," the ministry said in a statement.

"They wanted to provoke the servicemen into trying to destroy the kindergarten."

Ukraine's military launched an offensive against pro-Russian insurgents in mid-April, but has failed to oust them from their strongholds in more than a dozen towns and cities.