WARSAW — Europe’s right-wing populists have scrambled to outdo one another in celebrating Donald J. Trump as an American president who shares many of their nationalistic, anti-immigrant attitudes. Less thrilled are the hard-liners in Poland. And the problem is Vladimir V. Putin.

Poland already has one of Europe’s most conservative governments, which took office a year ago. And top leaders have eagerly cast Mr. Trump’s election as the latest and most crucial chapter in a global shift away from Western-style liberal democracy. But they also see a worrisome problem: the chummy noises Mr. Trump has made toward Mr. Putin, the Russian president.

Even as the government may share many of Mr. Putin’s conservative attitudes and nationalist impulses, Polish leaders are restrained by a deep, almost visceral distrust of him. Memories of Russian domination during the Soviet era are still raw here, and Poland has been one of the most hawkish members of the European Union on taking a hard line toward Mr. Putin.

The Polish predicament is yet another reminder of how thoroughly the Trump victory has scrambled the geopolitics of Europe. His campaign remarks undercutting NATO, and suggesting that the United States could possibly walk away from the trans-Atlantic alliance, alarmed leaders in the Baltics and in Poland — even after Mr. Trump softened his remarks.

Most of all, Poland and the rest of Europe will now wait and see whether the victory of Mr. Trump, who apparently has an affinity for Mr. Putin, is an opportunity for Europe, or an existential threat. Prime Minister Viktor Orban of Hungary, who had already made friendly gestures toward Moscow, sees opportunity. Many Poles still see risks.

“What Poland can do is keep increasing its military capabilities, keep a low profile and just wait for Putin to show his true colors,” said Jerzy Targalski, a right-wing political analyst and historian who supports the governing Law and Justice Party. “And it won’t take too long.”

For now, Polish leaders have a dual-track strategy to praise Mr. Trump’s victory as a continuation of their own victories, focus on areas of agreement with the new American president and play down, as much as possible, disagreements over things like how to deal with Moscow.

Poland’s prime minister, Beata Szydlo, joined other right-wing leaders in declaring Mr. Trump’s victory a validation of the rise of a more nationalistic and muscular form of government with a strong executive and resurgent patriotism.

“A certain era in world politics ends, an era when politics focused on elites and dealing with issues of the elites,” she said. “Democracy won despite the liberal propaganda.”

But at the same time, Poland’s president, Andrzej Duda, was quick to add that his country still expected the United States to abide by its NATO commitments and to proceed with plans to deploy NATO troops, including thousands of Americans, this spring along the European Union’s eastern flank.

Of course, even some members of Mr. Trump’s own Republican Party do not agree on the best way to approach Mr. Putin — whether to continue international pressure, as Poland wishes, or to seize the moment to develop a closer relationship that ends Russia’s isolation, as Poland fears.

Poland and the Baltic States, geographically closer to Moscow, have insisted on maintaining pressure. On Thursday, Lithuania’s Parliament voted almost unanimously — with just one abstention — to maintain economic sanctions as punishment for Russia’s annexation of Crimea and aggression in eastern Ukraine, for instance. On Friday, key European leaders, along with President Obama, pledged to renew those sanctions.

But other, more Putin-friendly leaders have indicated they might agree to end the sanctions. Presidents who were elected just this month in Bulgaria and Moldova had talked during their campaigns about doing just that, while political leaders and analysts in Poland and elsewhere expect Mr. Trump to move quickly to negotiate a new “grand bargain” with Mr. Putin.

They suspect Mr. Putin also wants to move swiftly, not only to end the sanctions that are squeezing his country’s economy but also, perhaps, to head off that planned spring deployment of NATO troops.

Polish officials have preferred to express confidence that, despite the troubling signs, the new president in the end will abide by America’s NATO and European Union commitments.

“I think we can be confident that U.S. policy toward Poland and thus also our relations will not deteriorate, that at least they will not become weaker,” Mr. Duda said after a brief phone conversation with Mr. Trump this past week. “And maybe we will be able to strengthen them even further.”

Polish leaders have focused on the more welcome statements Mr. Trump has made, like his promise to try to change rules that force Poles to get visas to visit the United States, long a sore point. And they have steadfastly defended Mr. Trump from criticism in more liberal corners of Europe.

As a result, the right-wing government — which has chafed under international criticism that its moves to marginalize the country’s constitutional court amounted to an undemocratic attack on the rule of law — hopes that Mr. Trump’s victory will create an atmosphere in which it can resolve that crisis more to Warsaw’s liking.

“The problem is not that there is a perception that Poland is under immediate threat from Russia,” said Marek Menkiszak, head of the Russian section at the Center for Eastern Studies in Warsaw, an independent but government-funded research group. “The problem is rather this uncertainty caused by the mixed signals that have come out of the Trump team.”

Rudolph W. Giuliani assured a gathering of Polish-American voters that the Trump administration would abide by its treaty obligations, including going to the military aid of any NATO member state that is attacked. But then Newt Gingrich, also acting as a Trump surrogate, remarked that if he were president he would not necessarily go to the aid of Estonia, which he described as essentially a suburb of St. Petersburg, sending a shiver down the spine of many Eastern European leaders.

And the candidate himself alternated between telling Poland that America would stand behind its ally and making comments about NATO having, perhaps, outlived its usefulness.

Optimists point out that pretty much every recent American president went into office hoping to refresh relations with Moscow, only to see those efforts stall.

“George W. Bush and Obama also announced a policy of rapprochement with Russia,” said Peter Kreko, the director of the Political Capital Institute, a research group in Budapest. “But in the end, none of them succeeded.”

Polish officials and analysts agree, saying they hope to cement strong relations with the Trump administration — stressing shared concerns like immigration and national sovereignty — and hope Mr. Trump does not surrender too much, in the short term, in his desire for a deal with Mr. Putin.

“You can’t make a deal with Russia and then expect it to keep the terms of the agreement,” said Mr. Targalski, the political analyst. “It’s never a real partnership with Russia. It will take everything the U.S. is willing to offer, and then just keep on testing the new administration’s patience.”

The underlying hope, Polish officials and analysts say, is that the Donald J. Trump of the campaign — praising Mr. Putin, belittling NATO, bashing trade treaties — will be different from the Donald J. Trump in the Oval Office. Any weakening of the trans-Atlantic alliance would have a huge impact on Poland and the region, they say.

“Being in this part of Europe, you depend on alliances,” said Marcin Zaborowski, executive vice president the head of the Warsaw office of the Center for European Policy Analysis. “Without alliances, we would become the Europe of the 1930s, which did not end too well.”