The U.S. has seen no evidence that Russia has begun its announced withdrawal of troops from the Ukraine border area, a Pentagon spokesman said Tuesday.

"President Putin said he's ordered them back to their home bases, which to us means a wholesale withdrawal of all the forces that are arrayed on the Ukrainian border. We've not seen that yet," Navy Rear Adm. John Kirby said.

Russian President Vladimir Putin shakes hands with parliament member Valentina Matviyenko at a meeting in the Russian Black Sea resort of Sochi on Monday, May 19, 2014. (AP Photo/RIA Novosti Kremlin, Mikhail Klimentyev, Presidential Press Service)

"They are still in the tens of thousands," Kirby said. "We have not seen any withdrawal activity." NATO officials have estimated that 40,000 Russian troops are arrayed along the border with Ukraine.

Kirby said small numbers of Russian troops have gone back and forth to forward operating bases near the border.

Russia's Defense Ministry has said army units were dismantling camps in the border region and preparing to leave. President Vladimir Putin ordered the retreat Monday in a move that appeared to be aimed at easing tensions with the West.

Such withdrawal activity, if confirmed, would ease fears that the Kremlin was positioned to invade eastern Ukraine.

Kirby also announced that the USS Vella Gulf, a cruiser, would enter the Black Sea later this week as part of a series of U.S. military moves designed to reassure European allies of an American commitment to their security amid tensions over Ukraine.