Russian President Vladimir Putin has ordered troops deployed in regions near Ukraine to return to their home bases, the Kremlin said Monday.

The move appears to indicate Putin's intention to de-escalate the crisis over Ukraine, the worst in Russia's relations with the West since the end of the Cold War.

But NATO Secretary General Anders Fogh Rasmussen told reporters Monday the military alliance has "not seen any evidence at all that the Russians have started withdrawal of troops from the Ukrainian borders."

He said that NATO remains open to a political dialogue with Moscow, and has proposed holding a meeting at the NATO-Russia Council next week.

If we, one day, see clear evidence of a meaningful Russian withdrawal of troops from the Ukrainian borders, I would be the very first to welcome it, because it would be a step in the right direction.

The West has protested the deployment of 40,000 Russian troops near the border with Ukraine, seeing it as a possible preparation for grabbing more land after the annexation of Crimea in March.

Putin has made similar commitments in recent weeks concerning his troops in the region.

Putin has previously said he has ordered troops to return from the area near the Ukraine border, but the United States and NATO said they saw no sign of a pullout and have threatened more sanctions if Russia tries to derail Ukraine's presidential vote set for Sunday.

The Russian Defense Ministry insisted that there were no buildup near the border, saying that the troops in the regions in western Russia are involved in regular training.

Putin went one step further Monday, ordering Defense Minister Sergei Shoigu to pull out forces involved in such training in the Rostov, Belgorod and Bryansk regions, according to a statement released by the Kremlin.

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NATO rebuts Russia claim