Satellite images showed a long build-up of armoured vehicles and artillery starting 29 kilometres (18 miles) north of the city, as Moscow defied mounting global pressure and a wave of international sanctions that have smashed Russia's economy.
Indian foreign ministry spokesman Arindam Bagchi said the ambassadors of Russia and Ukraine had been called in "to reiterate our demand for urgent safe passage for Indian nationals who are still in Kharkiv and cities in other conflict zones."
Chubais was one of the principal architects of Boris Yeltsin's economic reforms of the 1990s and was Putin's boss in the future president's first Kremlin job. He later ran big state businesses under Putin and held political jobs, lately serving as Kremlin special envoy to international organisations.
Tens of thousands of residents have already fled the besieged southern port city, bringing harrowing testimony of a "freezing hellscape riddled with dead bodies and destroyed buildings", according to Human Rights Watch.
Hundreds of thousands are believed to be trapped inside buildings, with no access to food, water, power or heat. Both civilians and Ukrainian troops were coming under Russian fire, said regional governor Pavlo Kyrylenko.
The bi-annual exercise, known as Cold Response, has taken on an added significance for some of the roughly 3,000 US Marines taking part in it because of Russia's invasion of Ukraine.
Russia is facing an onslaught of international sanctions led by Western nations aiming to isolate it from the global economy, including notably shutting it out of the SWIFT global bank messaging system and restricting dealings by its central bank.
The United States imposed new sanctions Thursday on entities and people in Russia and North Korea after Pyongyang's latest ICBM missile test.
The action agreed on Thursday comes as the alliance says it will double the number of battle groups defending the alliance’s eastern border in light of the Russian invasion.