Yet in an indication of how seriously Brussels takes the current standoff with Belarus, Charles Michel, the president of the European Council, visited Warsaw on Wednesday to meet with Prime Minister Mateusz Morawiecki of Poland to offer solidarity — and even, perhaps, some border funds.

“Poland, which is facing a serious crisis, should enjoy solidarity and unity of the whole European Union,” Mr. Michel said. “It is a hybrid attack, a brutal attack, a violent attack and a shameful attack,” he added. “And in the wake of such measures, the only response is to act in a decisive manner, with unity, in line with our core values.”

Chancellor Angela Merkel of Germany called President Vladimir V. Putin of Russia, urging him to push Belarus to stop its “inhuman and unacceptable” actions at the Polish border, her spokesman said.

Moscow supports Mr. Lukashenko with money and personnel. Unsurprisingly, the Kremlin said, Mr. Putin told Ms. Merkel that there was nothing he could do and that the European Union should deal directly with Mr. Lukashenko. Which is exactly what Brussels refuses to do.

But the position of Brussels is delicate, presenting the European Union with a three-pronged problem. It must show solidarity about protecting the borders of the bloc, sympathy about the humanitarian crisis unfolding there and firmness about defending the supremacy of European law.

The Europeans can hardly ignore the sight of innocent children, women and men, however manipulated they may have been, in freezing conditions, stuck between Polish border guards and troops and barbed wire, and Belarusian troops. The soldiers will not only prohibit them from returning to Minsk, the Belarusian capital where many are arriving before moving to the border, but are also actively helping them breach the Polish border.

At least 10 people have already died; other estimates are higher, but Poland has barred journalists and nongovernmental organizations from the border area.

In response, Brussels is contemplating a fifth round of sanctions, perhaps as early as Monday, aimed at Belarusian officials and at airlines that are flying migrants from the Middle East to Minsk. But few believe that new sanctions will move Mr. Lukashenko any more than previous ones have done, especially since his efforts are a response to the sanctions already in place.

“This is a very serious crisis for the European Union, not just for Poland,” said Piotr Buras, a Warsaw-based fellow of the European Council on Foreign Relations. “It’s a crisis of security, which could get much worse if Polish and Belarusian guards start shooting, and it’s a very serious humanitarian crisis, because Europe can’t accept people starving and freezing on the border.”

Given the nature of the crisis, Mr. Buras said, Brussels should separate it from the confrontation over the rule of law: “Whatever we may think about the Polish rule of law crisis, the E.U. must act in its own interest.”

But the Polish government, which no longer has a clear majority in Parliament, is itself politically stuck, Mr. Buras said. “The problem is not that the E.U. doesn’t want to help Poland because of the rule of law,” he added. “It’s the other way around — it’s very difficult for this Polish government to accept help from E.U. institutions that they are fighting on another front. And the government wants to present itself as the sole savior and defender of the Polish people.”

The European Union has offered Poland help with its own border guards, known as Frontex, significantly expanded since the 2015 crisis and based in Warsaw, said Camino Mortera-Martinez, a Brussels-based fellow of the Center for European Reform. And Brussels also has asylum support staff members who can help screen migrants to judge their qualifications for asylum.

But Poland has rejected both offers and insists on keeping the border area sealed. One reason is its fight with Brussels and its unwillingness to accept help. Warsaw also does not want the oversight of its actions that Frontex might provide, said Luigi Scazzieri, a research fellow in London who is also at the Center for European Reform.

Nor do Warsaw or Brussels want a screening procedure that will act as a “pull factor” to give Mr. Lukashenko and more migrants the hope that they can get into Europe this way.

“The concern on the government side, and this is why they’re so firm, is that if there is even a process to let people in, this will create a narrative that this is a place where people from Iraq and Syria can be processed into Europe, and the numbers won’t be 4,000, as now, but 30,000,” said Michal Baranowski, the director of the Warsaw office of the German Marshall Fund.

So policymakers are in a real conundrum for now, Mr. Scazzieri said. In the longer run, he suggested that sanctions against the airlines would reduce the numbers of migrants, and if the borders remained closed and were reinforced further, fewer would risk the journey.

And at some point, he said, Mr. Lukashenko “will understand that too many migrants in Belarus will create domestic problems.”

Monika Pronczuk contributed reporting from Brussels, and Anton Troianovski from Moscow.