They share an enthusiasm for sports, fitness and the occasional cigar. They are both unpopular leaders, scarred by terrorism and zealous in their warnings about the threat of Islamic extremism. And yet they profess grand ambitions to accomplish what other leaders have failed to do for decades: make peace between the Israelis and Palestinians.

President Bush and Prime Minister Ehud Olmert of Israel have in two years forged the sort of empathetic relationship that Mr. Bush had with the former prime minister, Ariel Sharon, and one that many in Israel and the United States thought unlikely to be repeated when Mr. Olmert came to power.

On Wednesday, as Mr. Bush arrived in Israel for his first visit as president, the bond between the men was clearer than ever. And it is the strength of their trust in each other, especially Mr. Olmert’s faith in Mr. Bush’s commitment to Israel’s security, that many here say may offer the best foundation for an agreement with the Palestinians before the end of Mr. Bush’s term.

“We certainly don’t want to delay the negotiation process,” Mr. Olmert said, “when we have such political assistance, assistance with respect to our security, too, when it comes to the most important power in the world being led by a person who is so deeply committed to the security of the state of Israel and to realizing the vision of two states, a person who is fair, who does not hide his viewpoints, who speaks openly about his will to establish a Palestinian state alongside Israel.”

Mr. Bush was here on Wednesday, and will go to the Palestinian territories on Thursday, to push Mr. Olmert and the Palestinian president, Mahmoud Abbas, to get serious about their negotiations and their obligations to each other, as written in the first stage of the “road map” outlining steps to be taken.

Appearing with Mr. Olmert after a day of pomp including an English and Hebrew version of “Over the Rainbow” sung by Israeli youngsters, Mr. Bush declared this a “historic moment, a historic opportunity” to overcome the deep skepticism here and elsewhere that the peace efforts begun in Annapolis, Md., in November would succeed.

“I’m under no illusions,” Mr. Bush said after two and a half hours of meetings, including an hour privately with Mr. Olmert. “It’s going to be hard work. I fully understand that there’s going to be some painful political compromises. I fully understand that there’s going to be some tough negotiations, and the role of the United States is to help in those negotiations.

“It’s essential that people understand, America cannot dictate the terms of what a state will look like,” he added. “The only way to have lasting peace, the only way for an agreement to mean anything is for the two parties to come together and make the difficult choices, but we’ll help and we want to help.”

In interviews before and during Mr. Bush’s visit, officials described the evolution of the deep bond between the leaders, reinforced by their shared views of Israel’s security, and their own political problems in selling their approach to their respective constituencies.

Mr. Bush’s relationship with the two Israeli leaders he has known best, Mr. Sharon and Mr. Olmert, have differed in detail, if not in spirit. Mr. Bush admired Mr. Sharon as “an old warrior” who took him, when he was governor of Texas in 1998, on a helicopter ride over the settlements and battlefields that crystallized Mr. Bush’s sympathies for Israel’s security concerns, a senior official who worked for both Israeli leaders said.

“With Olmert, it’s completely different,” the Israeli official said, speaking on condition of anonymity because he was discussing private interactions between the leaders. “They’re the same age. They’re both runners. They both feel that most of the world is against them, which, I think, is not far from the truth.”

Mr. Bush often relies on the personal in his foreign policy, responding to world leaders based on his own gut sense of their trustworthiness, as he expressly and, some say, wrongly did with President Vladimir V. Putin of Russia.

In this case, the men’s friendship was cemented during Mr. Olmert’s first visit as prime minister to Washington in May 2006. They sat on the Truman balcony at the White House, without aides, and smoked cigars. They talked for more than an hour about family and sports and not, the Israeli official said, about politics.

Their relationship is politically useful to both of them, as both seek, in their own ways, to shore up their legacies as leaders. A large photograph of Mr. Bush and Mr. Olmert, walking shoulder to shoulder, hangs prominently in the West Wing of the White House. In Mr. Olmert’s private study, there are two photos of him with Mr. Bush, one like the one in the White House and the other with Mr. Olmert’s hand on the president’s shoulder.

For Michael Oren, an Israeli historian of American-Israeli relations, “the message is very clear” that Mr. Bush is a strong supporter of Israel and of its current prime minister.

However warm, the relationship is not one of equals. “They have a strong personal rapport,” said Miri Eisin, who just left the job as Mr. Olmert’s spokeswoman. “But in the end, Bush is the leader of the free world, someone whose decisions affect the entire world. And you see the dynamics of that in the room.”

Mr. Olmert’s effusive praise of Mr. Bush can embarrass Israelis, but they also understand that the relationship with Washington is central. Mr. Bush, as usual, is more circumspect in his public comments. Part of that may be personality, but part also reflects the power of Washington and the need to try to seem even-handed between the Israelis and the Palestinians.

Mr. Olmert, who is said to have begun his acquaintance with Mr. Bush with a little skepticism, fed by his dovish wife, Aliza, has come to admire and trust Mr. Bush, his aides say. They say he believes that Mr. Bush, with his post-9/11 stance against terrorism and his belief in Israel’s democratic values, is a dependable ally who understands Israel’s security problems, both with the Palestinians and regionally, with Iran, and who is committed to defending Israel’s existence.

For Mr. Olmert, the close connection to Mr. Bush is both a lifeline and an insurance policy, that Israel will not be pressed to sacrifice its security to satisfy the American desire for a peace treaty.

Greeting Mr. Bush on Wednesday, Mr. Olmert told him, “Since I took office two years ago, you have become my personal friend and confidant.”

In an interview with the Israeli newspaper Yediot Aharanot, Mr. Bush praised Mr. Olmert as a man with a vision. “I trust him, I like him, and I think he’s a man of strength,” the president said.

Mr. Bush was said to have admired Mr. Sharon, incapacitated by a stroke, as a war hero and resilient politician, and to have treated him with respect. “With Olmert, there’s not the awe Bush had of Sharon as a great warrior, a little like Bush’s father,” Mr. Oren said.

Mr. Sharon also infuriated Mr. Bush at times, once by indirectly comparing him, in 2001, to Neville Chamberlain when he warned Mr. Bush not to appease Arab nations the way that “enlightened democracies in Europe” appeased Hitler in 1938 by sacrificing Czechoslovakia.

Daniel Levy, an Israeli analyst with the New American Foundation in Washington, said Mr. Bush and Mr. Olmert had grown so close that the president was now invested in his political future, willing to visit Israel so soon after Annapolis at least in part to bolster his standing before the Winograd report on the Lebanon war is made public later this month.

“He’ll make sure he knows the extracurricular interest of his interlocutor,” Mr. Levy said. He called it “an act of fidelity to Olmert.”

Their exchanges of gifts were also telling. Mr. Bush gave Mr. Olmert, a soccer fanatic, a soccer ball, a sports bag and cufflinks. Mr. Olmert gave Mr. Bush, who has traded running for biking, an Israeli national bike-team uniform, a water canteen emblazoned with “George W. 43” and a global positioning system for the handlebars, loaded with the trails on his Texas ranch and riding paths in Israel.

When the G.P.S. is turned on, the American and Israeli flags appear, and the sentence: “To my friend George Bush, from one athlete to another, happy trails.”