The official, speaking on condition of anonymity, said Russian troops "continue their repositioning and to move off the border."
Russian President Vladimir Putin, who ordered the deployment of close to 40,000 troops on the Russia-Ukraine border in March, according to NATO, said on May 19 the soldiers had been recalled to their garrisons.
Estimating the Russian drawdown in the "thousands," the US official added that "tens of thousands remained massed on the border.
Ukraine's army said today it had regained control of the airport serving Donetsk in eastern Ukraine after fierce fighting with pro-Russian militants.
Putin, accused of supporting the separatists, demanded that Ukrainian authorities immediately halt operations against the militants.
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