PARIS – European allies alienated by President Bush’s first four years in power offered yesterday to let bygones be bygones, saying they want to work with the administration and seeking, right from Day 1, to get the White House to listen more to overseas opinion. President Chirac, in a congratulatory letter, said he hoped Mr. Bush’s second term “will be the occasion for strengthening the French-American friendship.”

“We will be unable to find satisfying responses to the numerous challenges that confront us today without a close trans-Atlantic partnership,” wrote Mr. Chirac. He addressed the letter to “Dear George.”

Another critic of the Iraq war, the prime minister of Spain, Jose Luis Rodriguez Zapatero, said his government wants “a relationship of efficient, constructive cooperation with the U.S. government and with President Bush, respecting the ideas of each side.”

Mr. Zapatero, who angered Washington by withdrawing Spanish troops from Iraq, stayed up most of the night to watch as Republican red crept across the American electoral map. Election interest in Europe was intense, as was the disappointment many felt over Mr. Bush’s victory.

“There is a major and lasting lack of understanding between the American people and the rest of the world, in both directions,” said Hubert Vedrine, a former French foreign minister.

Others worried that Mr. Bush, strengthened by a bigger win than in 2000 and backed by a Republican Congress, would turn a deaf ear to world concerns. “Europe will continue to criticize Bush the same way as earlier,” said the prime minister of Sweden, Goeran Persson. “But I do not believe that he will be more willing to listen.”

The German Foreign Ministry’s top official for relations with Washington, Karsten Voigt, said he hoped Mr. Bush would seize the chance for “a new beginning” with Europe.

The American leader would do well to “approach the Europeans…and say, let us sit down and talk about where we have common interests,” Mr. Voigt said. Although Germany has offered to help rebuild Iraq and is training new Iraqi police and military, Mr. Voigt said “the German military will not be sent to Iraq.”

Bush allies in the war on terror took comfort in continuity.

“From our point of view, the Bush administration is a known quantity,” said the Australian foreign minister, Alexander Downer. “We’ve had a very good relationship with them for the last four years and I’m sure we’ll be able to keep building on that over the next four.”

President Putin said a Bush victory would mean the American people had not given in to terrorist threats. “I would feel happy that the American people have not allowed themselves to be scared and made the decision they considered reasonable,” Mr. Putin said at a Kremlin news conference after talks with the prime minister of Italy, Silvio Berlusconi. “Bush will keep up that policy that gives the United States the role of promoting freedom in the world,” Mr. Berlusconi said.

Prime Minister Blair pledged to work with Bush in the war on terrorism and in revitalizing the Middle East peace process, and called on Europe and America to “build anew their alliance” after strains created by the Iraq war.