ASHINGTON — Back in 2001, while preparing for his first trip across the Atlantic, President Bush startled some visitors with a question: Was the drive toward European unity good for the United States?

Then he smiled, as if to say he was just trying to learn more about a complex subject, a guest recalled.

But now on the eve of Mr. Bush's first trip to Europe since re-election, the question hangs pungently in the air.

Mr. Bush will go to Brussels, the capital of the new Europe, with a message that whatever bad feelings remain after the war in Iraq the United States embraces a newly integrated Europe and would not dream of sowing divisions for its own purposes.

Only a year and a half ago, though, The Weekly Standard, a conservative magazine with close ties to administration hawks, was arguing that the United States should oppose a united Europe because it would inevitably succumb to the supposedly weak-kneed policies of France and Germany.

"Can anyone now seriously believe that a single E.U. foreign policy will be more helpful to the United States than a British, a Polish, and a Spanish one?" it asked.

With that question, the magazine got to the heart of an argument that remains alive in at least parts of Washington.

The case for the overriding importance of a united Europe to Mr. Bush's goals is in many ways obvious. Although the United States proved it could win the Iraq war by itself, it has found it cannot win the peace unless the world - and Europe in particular - blesses Iraq's transition to democracy.

Nor can it hope to lead Israel to the peace table without the Europeans leading the Palestinians there, too. Cooperation between the world's two biggest economic blocs, on everything from terrorism to natural disasters, is clearly critical for progress.

Simply put, if a united Europe can agree with the United States, the two acting together are much more likely to succeed.

After all, the 25-nation European Union weighs in with $11 trillion in economic output, roughly on par with the United States, and a currency so strong that the euro may one day replace the dollar as the principal means of exchange. And as a whole, the most turbulent Arab societies, including the Palestinians, trust the European Union's emissaries to look after their interests more than they trust President Bush.All of that said, some fear remains in parts of the Bush administration - like the Defense Department and the office of Vice President Dick Cheney - that this new Europe is not always in tune with American goals. Europe has, in this view, become lethargic, self-absorbed, even pacifist about protecting its interests.

Some conservatives see this new Europe as focusing on building itself up as a counterweight to American interests, instead of being vigilant against new threats.

One focus of American concern is the kind of talk that was heard at a recent security conference of Europeans and Americans in Germany - suggestions that Europeans want to downgrade the trans-Atlantic partnership of NATO in favor of a European Union with an independent military capability. That sort of talk enrages Bush administration officials, who see it as an attempt to lessen American say in how Europe uses its muscle.

While it may be true that no American administration has ever set a policy of wanting a divided Europe, skepticism about a unified Europe with its own military is neither new nor likely to disappear.

"A lot of the language heard now about an independent European Union force was first invented by a very wary Clinton administration," said Robert Kagan, a conservative author who is now at the Carnegie Endowment for International Peace. "America has always been of two minds about a united Europe. There's the Acheson model of needing a united Europe to keep the Germans in the game without alarming everyone, but we've always had our concern about Europe returning as a great power."

Mr. Kagan notes, however, that the United States won't be able to stop Europe from elevating the European Union over NATO if it wants to. "We can't prevent it from happening," he said. "We need to get over it and deal with it."

To some degree, Europeans have already welcomed the message that the United States has moved beyond the discussion of whether a united Europe is good or bad for Americans, in part because Secretary of State Condoleezza Rice went out of her way to assure Europe about American intentions when she visited on her own earlier this month.

"To me the debate in the United States is over, and it's very good that it's over," Javier Solana, the European Union's high representative for foreign and security policy, said in an interview. "We are hearing a very clear message that it is better to have a European ally that is strong and united, even if it's more troublesome at times, than to have someone who is weak and divided."

To drive home that point, Mr. Bush will travel not to Paris or Berlin or London, but to Brussels, headquarters of NATO, the European Commission and the European Union, to proclaim his desire to work together against terrorism and nuclear proliferation and for Palestinian-Israeli peace talks and democratic reforms.

Henry Kissinger, the former secretary of state, used to joke that he would take "Europe" seriously as an entity when it had a telephone number he could dial. Mr. Bush seems to recognize that the number at least has a Brussels area code.

Still, a feeling persists in Washington that the new Europe, impressive as it is, remains cumbersome and self-absorbed and that future crises may well require America to find allies where it can. On that score, some challenging examples are on the table for discussion - notably Iran and China.

In general, the United States wants to confront the government in Tehran over its suspected nuclear weapons development by threatening sanctions, hinting at future military actions and holding out hope that Iranians may throw out their current regime.

Europeans, in general, want instead to talk to Iran about economic and political incentives if it voluntarily abandons work that could produce a nuclear bomb. They are likely to appeal to Mr. Bush this week to join in such discussions or risk being blamed for their failure.

But there are subtle divisions in Europe, with Britain sympathizing more with the American desire for confrontation, while France is somewhat less sympathetic and Germany the least. The Germans, in fact, are the most eager for the United States to join the negotiations. "So far we've had expressions of support from Washington but not much actions of support," said Wolfgang Ischinger, the German ambassador here.

Similarly, on China, Europeans seem poised to lift an arms embargo, arguing that such a step is the best means to moderate China's behavior. The United States strenuously opposes this, saying it could dangerously tip the balance of power in East Asia. Here it is less clear whether there are differences in Europe to exploit, since European countries, across the board, are eager for contracts with the Chinese.

Few experts expect a convergence or consensus to emerge from Mr. Bush's discussion of these and other issues, but they say it would help if Europeans feel he is listening.

"If the president just comes to Europe and says, 'We love you, but here is what you have to do,' he's not going to convince Europeans of anything," said Timothy Garton Ash, professor of European studies at St. Antony's College at Oxford University. "He has to convey a sense of real conversation between two partners."

A final paradox of European-American tensions is this: The one American strategy most likely to unite Europe would be to try to divide it. That is because in a newly united and integrated Europe, former Communist countries like Poland have a much greater interest in keeping the social welfare, subsidies and economic boon they get by tying themselves to Europe than any advantage they might gain by heading off on a new military venture with the United States.

"It's not that Europeans think the United States wants a weakened, sluggish or divided Europe," said Tony Judt, the director of the Remarque Institute at New York University.

"The fear is that what American wants is a Europe that does its bidding," Mr. Judt said. "But to suppose that because there are divisions in Europe, American policy can exploit them and turn some countries into American allies would be a big mistake. It would just backfire."