The disclosure by the officials, speaking on condition of anonymity, adds to growing US concern that Russia could be preparing for a new invasion of Ukraine as it has amassed more than 100,000 troops near its borders.
The Russian Defence Ministry did not immediately respond to a written request for comment.
Russian President Vladimir Putin said the West has not addressed Moscow's main security demands in the crisis over the former Soviet state - but he said that it was ready to keep talking, offering some hope that an attack is not imminent.
Putin offered his first reaction to the US and NATO responses to Russia's demands in a phone call with French President Emmanuel Macron after weeks of personal public silence.
The Kremlin quoted Putin as telling Macron he would study the responses provided by Washington and NATO this week before deciding on further action.
Current and former US officials said indicators like the blood supplies were critical in determining whether Moscow would be prepared to carry out an invasion, if Putin decided to do so.
A French presidency official said Putin, in his call with Macron, had underlined that he did not want the situation to intensify, echoing conciliatory comments by Russian Foreign Minister Sergei Lavrov, who said Moscow did not want war.
"Attention was drawn to the fact that the US and NATO replies did not take into account Russia's principal concerns," the Kremlin said of Putin's conversation with Macron.
It listed those concerns as avoiding NATO expansion, not deploying offensive weapons near Russia's borders and returning NATO "military capabilities and infrastructure" to how they were before former Warsaw Pact states in eastern Europe joined.
"The key question was ignored - how the United States and its allies intend to follow the principle of security integrity ... that no one should strengthen their security at the expense of another country's security," it said.
The United States and NATO have said some of Russia's demands are non-starters but have also left the door open to dialogue.
NATO Secretary-General Jens Stoltenberg said the Western military alliance was watching closely as Russia moves troops and arms into Belarus for drills.
He said NATO was ready to increase its troop presence in eastern Europe if Russia took further aggressive actions against Ukraine, and cautioned that a Russian attack could take many forms including a cyber attack, attempted coup or sabotage.
"From the NATO side we are ready to engage in political dialogue. But we're also ready to respond if Russia chooses an armed conflict confrontation," Stoltenberg said in Brussels.
US Defense Secretary Lloyd Austin said the United States remained focused on countering Russian disinformation, including anything that could be used as a pretext for attacks on Ukraine.
"While we don't believe that President Putin has made a final decision to use these forces against Ukraine, he clearly now has that capability," Austin told reporters.
He said "there are multiple options available to (Putin) including the seizure of cities and significant territories, but also coercive acts and provocative political acts like the recognition of breakaway territories."
US General Mark Milley, chairman of the Joint Chiefs of Staff, said Russia's deployment was larger than anything in scale and scope since the Cold War and said the United States strongly recommended that Russia stand down.
If Russia invades, the outcome would be "horrific" and result in significant casualties, Milley said.
Lavrov said he expected to meet US Secretary of State Antony Blinken again in the next couple of weeks. Their last meeting, in Geneva on Jan. 21, produced no breakthrough.
"If it depends on Russia, then there will be no war. We don't want wars. But we also won't allow our interests to be rudely trampled, to be ignored," Lavrov told Russian radio stations.
He said, without giving details, that the US counter-proposals were better than NATO's.
A senior US administration official said the United States welcomed Lavrov's comment on Russia not wanting war, but "we need to see it backed up by swift action".
A U.N. Security Council meeting on Monday on the build-up of Russian forces on the border with Ukraine will be "an opportunity for Russia to explain what it is doing", the official said, speaking on condition of anonymity.
The United States and the European Union have warned Russia that it will face economic sanctions if it attacks Ukraine.
These would build on sanctions imposed on Russia since it annexed Crimea and backed separatists in east Ukraine in 2014, though there are divisions among Western countries over how to respond as Europe is dependent on Russia for energy supplies.
US President Joe Biden and Ursula von der Leyen, who heads the EU executive, the European Commission, said they had agreed to cooperate on guaranteeing Europe's energy security but gave no details.
Washington has been in talks with energy-producing countries and companies around the world over a potential diversion of supplies to Europe if Russia invades Ukraine, a senior Biden administration official told reporters this week.
EU officials have repeatedly called for unity in the bloc over Ukraine, with some concerned that Germany - worried about energy supplies - has not taken a tougher stance.
Russia has dismissed calls to withdraw its troops, saying it can deploy them as it sees fit on its own territory. It has cited the Western response as evidence that Russia is the target, not the instigator, of aggression.
Ukraine has suggested a Russian attack is not imminent though an economically damaging war is possible. President Volodymyr Zelenskiy criticised a "feeling abroad" that a war had already started.
"We don't need this panic,"
Zelenskiy told reporters in Kyiv.