* Defence Ministry says troops begin leaving border areas
* NATO, United States say no sign yet of withdrawal (Adds White House, Pentagon comments, paragraphs 1, 7-9)
By Steve Gutterman
MOSCOW, May 21 (Reuters) - Russia said on Wednesday that troops deployed for exercises near the Ukrainian border had dismantled equipment and were moving to train stations and airfields for return to their permanent bases, but the United States and NATO said they saw no clear signs of a pullout.
The Kremlin said on Monday that President Vladimir Putin had told his defence chief to order troops to pull back from the frontier with Ukraine, where eastern regions have fallen largely under the control of pro-Russian rebels.
After spending a day dismantling field camps, packing and preparing military vehicles, forces in the Rostov, Belgorod and Bryansk provinces "have begun to move toward train stations and airfields", the Defence Ministry said in a statement.
It said troops were returning to their permanent bases, but did not specify how many troops were leaving the area and made no mention of two other provinces that border eastern Ukraine.
A withdrawal, cooling Western fears of a any immediate Russian intervention, could ease tension before Sunday's presidential election in Ukraine. The United States and EU hope the vote will strengthen the central government in Kiev, which is fighting pro-Moscow separatists in the east.
NATO has said Russia had amassed some 40,000 troops near the border, adding to tension since Russia's annexation of the Crimea region which brought relations to a post-Cold War low.
The White House and the Pentagon acknowledged activity on the border but said it was too early to determine whether it pointed to a pullout, or just more repositioning.
"It's impossible at this early stage to tell whether or not this movement that we're seeing is simply more of the same or if this is preparations for a broader withdrawal," said Colonel Steve Warren, a Pentagon spokesman.
White House spokesman Jay Carney said: "Should this be the beginning of a withdrawal we would welcome such an effort."
NATO Secretary-General Anders Fogh Rasmussen said on Wednesday that the alliance had still seen no "visible evidence" of a Russian troop withdrawal from the border.
"I wake up every morning hoping to see a real and meaningful withdrawal of Russian troops, but I have to tell you that so far we have not seen any visible evidence of a withdrawal," Rasmussen told a news conference in Sarajevo.
SET THE CLOCK BACK
Putin has reserved the right to send the military into Ukraine to protect Russian-speakers who dominate in the east.
Russia's Rostov province borders the Ukrainian regions of Donetsk and Luhansk, where pro-Russian separatists have seized government buildings in several cities and held referendums on secession this month. Rebel leaders say some 80 percent voted for a break with Kiev.
The deputy chairman of NATO's Military Committee said even if there were a pullback relations with Russia had already been damaged by the Ukraine crisis.
"You can't quite set the clock back and pretend nothing happened in the last two months ... We will have to reset to a new reality," Lieutenant General Mark Schissler said.
"The world is now a different place because of what's gone on, and in particular the continent of Europe is a different place," Schissler said.
"NATO had an aspiration to have a Europe that was free, whole and at peace. At risk now is being whole; at risk is being at peace." (Additional reporting by Missy Ryan in Brussels, Phil Stewart and Steve Holland in Washington and Daria Sito-Sucic in Sarajevo; Editing by Alissa de Carbonnel, Robin Pomeroy and Mohammad Zargham)