WASHINGTON — Many of the nearly 5,000 troops from the 82nd Airborne Division who arrived in Poland last week are working with Polish forces to set up processing centers for tens of thousands of people, including Americans, who are expected to flee neighboring Ukraine if Russia attacks the country, U.S. military officials say.

The Biden administration has repeatedly said U.S. troops will not fight in Ukraine or rescue Americans trapped there by a Russian attack. But U.S. commanders and their counterparts in Poland have been preparing parts of several Polish military facilities and erecting tents for possible evacuees.

So far, American officials said, there have been few, if any, people who have sought to use the facilities. But a large-scale Russian assault on Ukraine could result in one million to five million refugees, with many of them pouring into Poland, Pentagon officials have estimated.

That could lead to the largest flood of refugees in Europe since nearly a million Syrian refugees arrived in 2015, a surge that had a profound impact on European politics by bolstering far-right parties.

“Poland could see tens of thousands of displaced Ukrainians and others flowing across its border trying to save themselves and their families from the scourge of war,” Defense Secretary Lloyd J. Austin III said Friday in Warsaw.

Asked about the threat of a clash between American and Russian troops spilling over from Ukraine, Mr. Austin sought to tamp fears of an incident that could escalate into a confrontation between the two powers.

“We will make sure that we do everything possible to protect our troops and our Polish partners so that there isn’t a spillover,” Mr. Austin said in an interview with ABC News that aired on Sunday.

“This is something that we’ll be on the lookout for and we’ll be thoughtful about making sure that we’ve taken the right steps to try to prevent that,” he said.

Polish officials said they were preparing for the worst. “We are ready to help all those who will be forced to leave Ukraine,” said Mariusz Blaszczak, the Polish defense minister, speaking in Warsaw alongside Mr. Austin. “We, as a nation that has so strongly and badly experienced World War II, we know what support is all about, and we are ready, of course, to support all those who need the support and who suffer because of such an aggression.”

Another 1,000 American troops — a Stryker squadron from the Army’s Second Cavalry Regiment — moved into Romania from Germany on Saturday. The Air Force has also sent more than a dozen additional fighter jets to Eastern Europe in recent days to bolster aerial defenses there.

The American reinforcements more than double the number of American ground troops in the two countries, to roughly 9,000 in Poland and nearly 2,000 in Romania.

U.S. officials estimate that Russia has amassed 190,000 troops in and around Ukraine, including in the eastern part of the country, where Ukrainian forces and Russian-backed separatists have been fighting for years.

Pentagon officials have warned that the Biden administration would send even more troops to NATO allies in Eastern Europe in the event of a Russian invasion of Ukraine. Poland and Romania are both NATO countries, but Ukraine is not a member of the alliance.

“The troops that we have added to the already 80,000 that are based in Europe are going to reassure our allies and our partners to deter aggression against the alliance to conduct some joint training,” John F. Kirby, the Pentagon’s chief spokesman, told reporters earlier this month.

Besides any symbolic value, the 82nd Airborne soldiers will be thrust into the challenging job of helping the Polish authorities manage possibly tens of thousands of people fleeing Ukraine.

“Certainly, assistance with evacuation flow is something that they could do, and could do quite well,” Mr. Kirby said. “And they’re going to be working with Polish authorities on what that looks like, and how they would handle that.”

The 82nd Airborne troops, from Fort Bragg, N.C., are expected to train with Polish forces at several locations in the country, said a U.S. military official, who spoke on the condition of anonymity to discuss troop movements.

The Pentagon announced earlier this month that 3,000 troops from the 82nd Airborne would join 1,700 soldiers from the unit who deployed earlier to Poland. They are led by Maj. Gen. Christopher Donahue, who was the ground commander for the evacuation at Kabul’s international airport in August.

The Pentagon has barred reporters from embedding with the newly arrived troops. “We’re just not at a point now where we are able to provide that kind of access,” Mr. Kirby said.

Mr. Kirby said that while some of the Army troops may operate in eastern Poland, near the Ukrainian border, one thing is clear: “There’s no intention, there’s no plan and there’s no approval to put these troops into Ukraine. They’re being sent to Poland. They’re going to stay in Poland.”

Earlier this month, Mr. Austin ordered 150 Florida Army National Guard soldiers who had been training Ukrainian troops to exit the country “out of an abundance of caution,” Mr. Kirby said. That leaves a few dozen Army Special Forces personnel and Marine Embassy guards in Lviv, where the U.S. diplomatic corps was recently moved, as the last American military presence in the country.

In addition, some 300 members of the 18th Airborne Corps headquarters, also from Fort Bragg, have arrived in Germany to oversee the reinforcements flowing into Eastern Europe. The 18th Airborne Corps and this headquarters unit are led by Lt. Gen. Michael E. Kurilla, whom President Biden has nominated to take over the military’s Central Command in April. And last Thursday, Mr. Austin said 125 Army troops in Germany would go to Bulgaria for training.

The Pentagon in recent days has also dispatched aerial reinforcements to Britain, Germany and Eastern Europe. The Air Force said F-35 fighters from Hill Air Force Base, Utah, deployed to Germany days after B-52 bombers from Minot Air Force Base, N.D., were sent to Britain in a previously planned mission.

Eight F-15s from Seymour Johnson Air Force Base, N.C., have flown to Lask Air Base in Poland to augment the eight American F-15s there that are usually stationed in Britain.

“The deployment of U.S. F-15s to Poland elevates the collective defense capabilities on NATO’s Eastern flank and the enhanced air policing mission,” Gen. Jeffrey Harrigian, the head of U.S. Air Forces in Europe and Africa, said in a statement.

Eight F-16s usually based in Spangdahlem, Germany, have also deployed to Fetesti Air Base, Romania.

The buildup of aerial forces in Eastern Europe carries risks. U.S. and Russian warplanes operating in the eastern Mediterranean Sea flew dangerously close to each other in three separate incidents earlier this month, including one in which the aircraft came within about five feet of each other, Pentagon officials said.

In addition to the deployments to Poland and Romania, the Pentagon has put 8,500 other troops in the United States on “high alert” for possible dispatch to Eastern Europe. Those troops would take part in a NATO response force that might soon be activated, military officials said.

The bulk of the troops being put on higher alert are active-duty ground troops, including combat brigades, and medical, aviation, transportation, intelligence and surveillance forces.