George W. Bush, the most unpopular U.S. president in Europe since the advent of modern polling, will attempt just that in a European trip this week aimed at improving trans-Atlantic relations.

That Bush is reaching out is as clear as his schedule, from meetings with the leaders of NATO and the European Union to separate sitdowns with the three leaders most critical of U.S. policy in Iraq - French President Jacques Chirac, German Chancellor Gerhard Schroeder and Russian President Vladimir Putin.

From both sides of the Atlantic, leaders are touting a fresh start, fueled in part by the successful holding of elections in Iraq, signs of renewed hope for settling the Israeli-Palestinian conflict and a new U.S. emphasis on diplomacy.

Yet this week's trip also underscores the sharp differences that remain on policies from Iran to China, and the risk of even deeper U.S.-European tensions in the years ahead.

When Bush enters European Union headquarters in Brussels on Tuesday, he will pass banners touting the Kyoto Protocol, which mandates global reductions in carbon dioxide emissions and other pollutants associated with global warming.

In his first term Bush rejected the protocol but it took effect last week all the same. So, too, did the International Criminal Court, up and running despite continued U.S. opposition.

On those and other issues, the 25 European Union nations speak with an increasingly unified voice, imposing on members uniform standards - from abolition of the death penalty to targets on deficit control and foreign aid - on which the United States is increasingly out of step.

Bush acknowledged the European perspectives at a news conference Thursday and addressed them in part.

''There is a concern in Europe, I suspect, that the only thing I care about is our national security,'' he said, in light of the Sept. 11, 2001, terrorist attacks. ''But we also care deeply about hunger and disease.''

Simon Serfaty, a senior fellow at the Center for Strategic and International Studies, compared present U.S.-European tensions to those that followed World War II. There was resistance on both sides of the Atlantic to the proposals of President Truman, from the doctrine of Communist containment to creation of NATO, but the consensus that eventually emerged lasted a generation.

Serfaty hailed Bush's efforts since November, from a post- election news conference that stressed Europe's importance to conciliatory language in his Inaugural and State of the Union addresses. ''Everything he's done the past three months reflects a new sensitivity to Europe,'' Serfaty said. ''He has learned how to spell 'EU.' ''

Serfaty cautioned that whether Bush can translate that better start into concrete achievements is far from clear. What is certain is that the twists and turns of his first term cannot be sustained.

''The relationship will either get much better or much worse in the next four years,'' Serfaty said. ''It won't stay the same.''

Warning of worse is Thomas Donnelly, a senior fellow at the American Enterprise Institute. He said a new confrontation looms over the lifting of a European embargo on arms sales to China - in place since the 1989 Tiananmen Square demonstrations - that would make the dispute over Iraq look like small potatoes.

Donnelly said it was ''disturbing'' that Bush has so far downplayed the issue, as did Secretary of State Condoleezza Rice during her tour of Europe last month. ''The administration seems more interested in having happy talk'' on this trip, he said, but the issue can't be evaded for long.

''The cliff we're near to going over will create tensions far beyond what we've seen so far,'' he said. Bush's trip, he added, ''is like the last party on the Titanic before we hit the iceberg.''

European leaders, for their part, are putting the best face possible on Bush's overture, and the prospects for a new beginning.

"Do we have a new start? Definitely,'' Klaus Gottwald, deputy chief of mission for the German embassy, told a forum Thursday at the American Enterprise Institute.

His country was a leading critic of the Iraq war. His chancellor, Schroeder, told a security conference in Munich earlier this month that NATO as currently structured falls short as a forum for U.S.-Europe strategic dialogue. Yet as Gottwald previewed Bush's trip, he stressed the positives, declaring that the two countries had ''clearly buried the hatchet'' on Iraq and were finding common ground on Afghanistan and the Israeli-Palestinian peace process.

On the China arms embargo, Gottwald insists that U.S. concerns are misplaced. ''The label 'lifting the arms embargo' is wrong,'' he said. ''This is really about replacing a non-binding resolution from 15 years ago,'' from a much smaller EU, with ''a strong code of conduct'' that would restrict the sale of sensitive military technology.

Philip Gordon, director of the Center on the United States and Europe at the Brookings Institution, acknowledged that European arms sales to China would be ''intolerable'' to many here and ''really does have the potential to blow up into another U.S.-European crisis.''

China arms sales was one of the items on a proposed ''compact'' that Gordon circulated, with the signatures of 61 foreign policy specialists and former officials from the United States and Europe. Under the proposal, the United States would acquiesce in lifting the arms embargo; Europe would commit to tough safeguards. Comparable compromises are proposed for bridging U.S.-European differences on Iran, global warming and the International Criminal Court.

The goal ''is to go beyond words and diplomacy and actually make policy commitments,'' Gordon said.

Bush aides anticipate a warm reception in Bratislava, capital of the Slovak Republic and the only place where he has scheduled an address to a mass audience. He will speak in the city's historic town square, paying tribute to the ''champions of freedom'' from 1989 who helped topple Soviet and Communist rule across central and eastern Europe.

The post-Communist governments of those countries largely supported U.S. policy on Iraq, unlike France and Germany, prompting Defense Secretary Donald Rumsfeld's much-quoted reference to ''new Europe'' versus ''old.'' Scholars of the region say the distinction was overstated, as demonstrated most recently by Polish and Hungarian plans to pull troops out of Iraq, and that even on deeper ''freedom'' issues there is more ambivalence in eastern Europe than Bush has usually acknowledged.

''People are tempted to escape from the many burdens of freedom,'' said Charles Gati, a specialist on eastern Europe at the School of Advanced International Studies at Johns Hopkins University.

''I don't deny the desire for freedom but there's another desire, too - the desire to be taken care of,'' he said. ''Paternalism is seen as an alternative to pluralism. I expect pluralism to prevail but it's having a hard time, so far.''

Bush can witness firsthand one reason eastern Europeans are gloomier when he meets in Bratislava with Russian President Vladimir Putin. Much bad has happened since 2001, when Bush met Putin in Slovenia and said he had found a soulmate - from Putin's heavy-handed interventions in Georgia and Ukraine to his crackdown on independent media and businesses at home.

In an interview with Slovak State Radio on Friday, Bush said he wouldn't flinch from confronting the Russian leader.

''I have a good relationship with President Putin,'' Bush said. ''That's important, because that then will give me a chance to say in private - to ask him why he's been making some of the decisions he's been making.''