“I’M only a phone call away,” President Bush told the Israeli and Palestinian leaders last week, after they set the ambitious goal of negotiating a peace treaty by the end of next year. But as they joined him in the White House Rose Garden before going their separate ways, Mr. Bush had a slightly different message for the pair.

“I wish you all the best,” he said — a send-off that did not exactly give the impression he was eager to pick up the phone.

Coming from a president who has spent seven years shying away from the Arab-Israeli conflict, the ambivalent signals were not entirely a surprise. Mr. Bush has been given the tag “C.E.O. president,” happier to delegate than to delve into pesky details. But last week’s conference at Annapolis thrust him front and center into a role he never wanted to play, that of Middle East peace broker.

The question now is how Mr. Bush will play it. Will he keep his distance (as his aides insist is the plan), and simply give the parties encouragement to work it out themselves? Or will he be bitten by the bug that has led so many American presidents to wonder if, just maybe, they could go down in history as the man who brought peace to the Middle East?

Foreign policy experts say that if the Palestinians and Israelis are to settle their differences, it will take substantial presidential involvement. It won’t be enough for Mr. Bush to leave the work to his secretary of state, Condoleezza Rice. Unless he outlines a framework for negotiations himself, articulating what the United States would accept, they say, the parties will never take the effort seriously.

Mr. Bush, however, “is one of the few presidents since there has been an Arab-Israeli peace process who seem to have willfully disengaged,” said Aaron David Miller, a longtime Middle East negotiator whose book on the search for peace, “The Much Too Promised Land,” will be published next year. “I don’t believe that the tug of history on this issue, which seems to have been so alluring to many of his predecessors, provides much of a compelling attraction for this president.”

Perhaps not, but Mr. Bush is feeling the tug of history — his own — and there are powerful geopolitical reasons, particularly the rising regional influence of Iran, to draw him into the fray. At the same time, despite the contention of his press secretary that “the president is not a gambler,” he has always been willing to take big risks.

“Bush regards himself as the guy who routinely threw the long pass successfully,” said Bruce Buchanan, an expert in presidential psychology at the University of Texas. So it is not so far-fetched to think that, in his waning days in office, Mr. Bush will plunge into what Mr. Miller calls “the 50-year-headache” — especially if he gets a whiff that it could work.

“I think this is a legacy that he knows very well would be monumental,” said Zbigniew Brzezinski, who was national security adviser to President Jimmy Carter. “He would outshine Carter, he would outshine Clinton, he would outshine his father.”

Mr. Bush’s national security adviser, Stephen Hadley, said in a speech last week that the president would not “impose a peace plan with his name on it” but would help the parties “build the confidence necessary to make the hard choices.”

Richard Haass, a former State Department official in Mr. Bush’s first term, who also worked for the president’s father, said that such an effort would not be enough. “He needs to spell out his views for what would make for a reasonable peace,” he said. “The president doesn’t have to get as involved as, say, Jimmy Carter or Bill Clinton, but the president needs to create the context in which the secretary of state operates. That’s where Mr. Bush erred at Annapolis.”

One compelling reason for deep involvement, experts say, is that the Israelis and Palestinians expect it.

“They’re going to have to make painful compromises,” said Edward Djerejian, who served as ambassador to Syria under Mr. Bush’s father and ambassador to Israel under Mr. Clinton. “And it’s much easier for them to go to their domestic constituencies and say, ‘Look, this is not my preferred option, but the president of the United States is asking me to do this.’ ”

That was the experience of Warren Christopher, Mr. Clinton’s first secretary of state, who was dispatched to the region by his boss in 1993. Those who worked for Mr. Clinton say he was convinced that, through the sheer force of his own personality, he could bring the Arabs and Israelis together, and he spent much of his presidency trying.

“He basically called and paved the way for me with Prime Minister Rabin, and it meant everything,” Mr. Christopher said, adding, “It was clear that he was not on the sidelines, he was in the game.”

Mr. Carter, who viewed tackling Middle East peace as a religious and moral imperative, was so much in the game that he holed up for nearly two weeks at Camp David with the leaders of Egypt and Israel, trying to broker a treaty. Two days before the end of the historic talks, the Egyptian president, Anwar el-Sadat, grew fed up and ordered his aides to pack their bags. Mr. Carter went to Mr. Sadat’s cabin.

“He said, ‘I have to talk to you alone,’ and he took Sadat to the bedroom and he said, ‘If you leave, it’s the end of our relationship, the end of the U.S.-Egyptian ability to work together, it’s probably the end of my presidency,’ ” Mr. Brzezinski recalled, adding that he saw this as a sign of “the degree to which he was the principal motor driving this whole thing.”

Mr. Sadat stayed. Egypt and Israel ended their differences, one of the few lasting achievements in the quest for Middle East peace. And Mr. Carter went on to win a Nobel Peace Prize.

Mr. Bush himself has sniffed what Mr. Miller calls “the fumes of greatness that hover over this problem.” After a 2003 summit in Aqaba, Jordan, where Mahmoud Abbas, the Palestinian leader, and Ariel Sharon, then Israel’s prime minister, pledged to take initial steps toward peace, Mr. Bush promised to “ride herd.”

“People who saw him after that saw him as a changed man,” said Martin S. Indyk, a former ambassador to Israel who is now head of the Saban Center for Middle East Policy at the Brookings Institution in Washington. But the effort dissolved in violence, and Mr. Bush was consumed by Iraq.

At Annapolis, Mr. Bush got a taste of life as a hands-on negotiator when he helped Mr. Abbas and Ehud Olmert, the Israeli prime minister, put the finishing touches on their “joint understanding.” Mr. Bush emerged with the document in his hands — reading glasses still perched on his nose — and announced the agreement before delivering his own speech.

“He is now much more committed than he was two days ago, because he has entered the arena personally,” Mr. Djerejian said. “He has gone from the rhetorical to the actual; he is now engaged, whether he likes it or not.”