(CNN) -- NATO urged Russia to keep its word this time and withdraw troops along Ukraine's eastern border, saying it has seen no signs of soldiers returning to their bases.
President Vladimir Putin ordered tens of thousands of troops near the Ukraine border to return to their bases, his spokesman Dmitry Peskov said Monday. The withdrawal has started, he said, and could take some time to finish.
But despite Moscow's assertion, there were no signs of the troops' withdrawal, NATO Secretary General Anders Fogh Rasmussen said hours later.
The Obama administration was skeptical about Putin's order, while Kiev said it's monitoring the area to ensure troops are returning to their permanent bases.
"As you know, we have heard that promise before and have yet to see any indication that it's been fulfilled," said Jay Carney, the White House spokesman.
Rasmussen reiterated the need for the withdrawal and said he would "be the first" to welcome it.
"I think it's the third Putin statement on withdrawal of Russian troops ... but so far we haven't seen any withdrawal at all," he said. "Withdrawal of Russian troops will be the first step to de-escalating the situation."
Ukraine's presidential election scheduled for Sunday is crucial, Rasmussen said, and Russia and Ukrainian separatists should practice restraint to ensure it goes on as planned.
"We urge the armed pro-Russian separatist groups to stop their illegal activities. ... Russia should stop their support for these armed groups," he said.
"Russia should demonstrate a clear will to let the presidential election to go forward. The presidential election constitutes the best chance to find a sustainable solution to the crisis in Ukraine."
Russia's 'significant forces'
Putin's decision to amass 40,000 troops along Ukraine's eastern border triggered fears of an invasion ahead of the election.
Moscow defended the troop buildup, saying it was a routine exercise that has since ended.
Two weeks ago, Putin said Russian troops had pulled away from Ukraine's border and were merely conducting "regular exercises at the test grounds." At that time, NATO and Western officials said they saw no sign of widespread troop withdrawals.
But a senior U.S. administration official disputed Russia's assertion that they were routine exercises
"The fact is that Russia has been maintaining significant forces in forward deployment areas along Ukraine's border," the official said. "They have not been conducting routine training activities. They've been up on the border in a menacing posture, and we've been concerned about this military buildup and have been consistently calling on Russia to remove its troops back to their home bases and end this threatening behavior."
The official said the White House is monitoring the issue for evidence of withdrawal.
"As you'll recall, they've made similar claims before. They made them at the end of March and didn't follow through, so we'll be tracking this closely over the course of today and the coming days, and we'll want to see clear, firm evidence of this move before we make any judgment," the official said Monday.
The United States, which along with other Western countries has sanctioned Russia for its disputed takeover of Crimea, has threatened additional punishment for Russia if it fails to pull its troops back from the border.
Tensions in the east
Russian officials have said they reserve the right to protect the interests of Russian citizens and Russian-language speakers in Ukraine's east, which traditionally leans toward Moscow.
Tensions in that region remain high, with ongoing reports of violence and growing human rights abuses. Anti-terrorism operations that started in April in the Donetsk and Luhansk regions have killed 24 Ukrainian servicemen so far, said Valentyn Nalyvaichenko, head of the Ukrainian Security Service. He did not provide any more details.
In one of the latest incidents, Russian separatists clashed with Ukrainian border guards Saturday after a separatist leader was detained at a checkpoint.
Valeriy Bolotov, the self-declared governor of a "people's republic" in Luhansk, was detained by security forces in Dovzhanskiy. Attackers freed him after a firefight, but he was wounded and went to Russia for medical treatment, separatist spokesman Vasiliy Nikitin said.
Over the weekend, the Russian Foreign Ministry accused Ukrainian troops of attacking Ukrainian citizens and questioned whether Sunday's scheduled elections could be held amid the chaos.
"Such punitive action against its own citizens shows the hypocrisy of the Kiev authorities," a ministry statement said, referring to an international pact agreed to last month that called for an end to violence.
CNN's Virginia Nicolaidis, Ed Payne, Victoria Butenko and Richard Greene contributed to this report.