Mr Zelenskiy has accused Russia of flexing its military muscle and held a series of calls with Western leaders to rally support for Ukraine amid the tensions.
Russia has rejected the claims, arguing it can deploy its troops wherever it deems necessary on its territory, and that they are not threatening anyone.
It has accused Ukrainian authorities of trying to stoke up tensions in the east for domestic policy purposes.
The White House is said to be "increasingly concerned by recent escalating Russian aggressions in eastern Ukraine", including Russian troop movements on Ukraine's border.
Press secretary Jen Psaki said: "Russia now has more troops on the border with Ukraine than at any time since 2014.
"Five Ukrainian soldiers have been killed this week alone. These are all deeply concerning signs."
Speaking at a panel discussion with political experts in Moscow, Dmitry Kozak, a Putin aide who serves as Russia's top negotiator with Kyiv, warned Ukraine against using military force to retake control of the eastern regions, where many residents have received Russian citizenship.
"The start of hostilities will the beginning of an end for Ukraine," he said. "It would mean shooting itself not just in the foot, but in the head."
Mr Zelenskiy has accused Russia of flexing its military muscle and held a series of calls with Western leaders to rally support for Ukraine amid the tensions.
Russia has rejected the claims, arguing it can deploy its troops wherever it deems necessary on its territory, and that they are not threatening anyone.
It has accused Ukrainian authorities of trying to stoke up tensions in the east for domestic policy purposes.
Sky News
