NATO to put more forces on standby amid fears of Russian attack on UkraineNATO to put more forces on standby amid fears of Russian attack on Ukraine

Amid deepening anxiety over a possible Russian invasion of Ukraine, the United States announced it's pulling out diplomats' families and some staff from its embassy in the country. Meanwhile, NATO announced it was putting extra forces on standby.

© Mauricio Campino/U.S. Air Force via Reuters Airman 1st Class Olabode Igandan palletizes ammunition, weapons and other equipment bound for Ukraine during a foreign military sales mission at Dover Air Force Base, in Delaware, January 21, 2022. Picture taken January 21, 2022.

As Russia continues to mass tens of thousands of troops close to Ukraine's borders, NATO said the alliance was sending a small number of ships and fighter jets to Eastern Europe to strengthen its "deterrence" presence there and reassure its eastern members,

Denmark is sending a frigate to the Baltic Sea and four F-16 warplanes to Lithuania. At the same time, France is ready to send troops to Romania under NATO command, and Spain is considering deploying fighter jets to Bulgaria, NATO said in a statement. The Netherlands has agreed to send two F-35 jets to Bulgaria and has put a ship and land-based forces on standby for a NATO response force, officials said.

"NATO will continue to take all necessary measures to protect and defend all Allies, including by reinforcing the eastern part of the Alliance," NATO's Secretary General Jens Stoltenberg said in a statement.

The NATO announcement coincided with a report in The New York Times that the Biden administration may be preparing to send up to 5,000 American troops to Eastern European members of the alliance.

The White House and the Pentagon have not confirmed the report, though the administration has previously said sending more U.S. troops to Eastern Europe is on the table if Russia attacks Ukraine.

NATO on Monday said the "United States has also made clear that it is considering increasing its military presence in the eastern part of the Alliance."

The steps to boost NATO's readiness came as the U.S. State Department announced Sunday it was ordering the families of its diplomats at its embassy in Ukraine's capital Kyiv to leave the country over security fears.

The State Department said it has also authorized non-emergency staff at the embassy to depart voluntarily.

The United Kingdom on Monday followed suit, with its Foreign Office saying some embassy staff and their dependents would be withdrawn "in response to the growing threat from Russia."

Ukraine's government criticized the U.S. evacuation calling them "premature" and "excessively cautious."

"While we respect right of foreign nations to ensure safety & security of their diplomatic missions, we believe such a step to be a premature one & an instance of excessive caution," Oleg Nikolenko, a spokesman for Ukraine's foreign ministry, tweeted.

© Sean Gallup/Getty Images Civilian participants receive basic combat training in a forest on Jan. 22, 2022 in Kyiv, Ukraine.

Ukrainian officials are unhappy with the message the evacuations send by suggesting that a Russian invasion could be imminent. In general, they are much more skeptical that Russia is planning to launch a major attack and worry that western countries risk helping Moscow by exaggerating the risk and spreading panic.

Privately, American officials acknowledge there is a gap between the Ukrainian and U.S. assessment of the level of threat. Speaking on condition of anonymity, a U.S. official told ABC News this weekend that Ukraine was "p----- off" over the evacuations.

A senior State Department official on Sunday insisted the embassy drawdown did not undermine America's commitment to Ukraine, saying they were just "prudent precautions" given the heightened fear of a Russian attack.

The official said the decision was "based on this military buildup, based on how we see these developments," calling it the "right moment."

Those leaving the embassy will do so on commercial flights, the State Department has said, indicating it is not an emergency evacuation.

The State Department were scarred by the chaotic evacuation of Afghanistan, where thousands of Americans were stranded after the sudden Taliban takeover there caught the U.S. off guard. Officials are anxious to avoid a similar situation in Ukraine, should the worst happen.

Russia has repeatedly insisted it has no intention of attacking Ukraine. However, its military buildup continues near Ukraine's eastern border and now in Belarus, where trainloads of Russian tanks and artillery have been arriving for joint exercises there.

© Gleb Garanich/Reuters A security guard speaks on a mobile phone outside the U.S. embassy in Kyiv, Ukraine, Jan. 24, 2022.

A top commander of Russian-controlled separatist rebels in eastern Ukraine on Monday also accused Ukraine's military of preparing to launch an offensive against the separatist areas.

The U.S. and Ukraine are concerned that a false claim of a Ukrainian offensive against the separatists could be used as a pretext for Russia to launch an invasion.

Eduard Basurin, the head of the militia of the separatists' self-declared 'People's Republic of Donetsk' (DNR), in local media warned it "firmly recommends the enemy to give up its criminal intentions," promising the Ukrainian army "will suffer irreparable damage, after which it will not be able to recover."

Ukraine's government has insisted it will not launch any offensive and there is no evidence Ukraine is preparing to.