In November, Mr. Obama won re-election over Mitt Romney, who had been the not-so-subtle favorite of Mr. Netanyahu. Then on Tuesday, Mr. Netanyahu stumbled in his own re-election bid, with his Likud Party holding enough seats in Parliament to keep him in office but falling far short of expectations in the face of surging centrist voters.

Still, there was no crowing at the White House, at least in public, as the returns flowed in from Israel. Administration officials on Wednesday were reluctant to comment on how Mr. Netanyahu’s setback may affect his relations with Mr. Obama, especially since the Israeli leader has not yet begun the work of cobbling together a governing coalition.

As they sifted through the implications, analysts said there was more than vindication for Mr. Obama in Israel’s new political landscape.

Mr. Netanyahu’s weakened position could set the stage for, if not a “reset,” to use the administration’s well-worn phrase, then an improvement in his ties with the president.

If, as some analysts expect, Mr. Netanyahu seeks to put together a center-right coalition that includes Yair Lapid, whose Yesh Atid party won 19 seats in the 120-seat Parliament, it could sand away the roughest edges of Mr. Netanyahu’s existing right-wing coalition.

Mr. Lapid could push a new government in directions that would ease longstanding sources of tension with Mr. Obama. For example, he is more interested in creating jobs and providing housing than in expanding construction of Jewish settlements in the West Bank, a recurring source of friction between Mr. Obama and Mr. Netanyahu.

With Ehud Barak, a hawkish former general, leaving the Defense Ministry, Mr. Netanyahu may be under less pressure to consider a unilateral strike on Iran over itsnuclear program. That would be a relief to the White House, which has had to plead with the Israelis for patience while it pursues a last-ditch diplomatic effort with Tehran.

“A weaker Bibi heading a government with some centrists was the best outcome the White House could have hoped for,” said Aaron David Miller, a longtime Middle East negotiator, using Mr. Netanyahu’s nickname. “It gives them a better chance to avoid war with the Iranian mullahs and preserve the chance of a peace with the Palestinians.”

The most optimistic outcome, Mr. Miller said, would be a kind of “odd couple” relationship between Mr. Obama and Mr. Netanyahu, in which they retain their differences over issues like settlements, but learn to manage them more skillfully.

That would be no small step, given the mutual suspicion that has suffused their relationship. White House officials alternately fumed and rolled their eyes during the presidential campaign when Mr. Netanyahu appeared to tilt toward Mr. Romney, inviting him to dinner at his home during the Republican candidate’s visit to Jerusalem last July.

Mr. Obama, members of the Likud Party believe, returned the favor during the Israeli election when Jeffrey Goldberg, an American journalist who writes frequently about Israel, reported that the president had disparaged Mr. Netanyahu after the Israeli government announced plans for settlements in a contested area of the West Bank known as E1.

Mr. Goldberg quoted Mr. Obama as saying repeatedly, “Israel doesn’t know what its own best interests are.” The White House did not confirm or deny Mr. Obama’s comments.

But days before the election, Mr. Netanyahu shot back that “only Israeli citizens will be the ones who determine who faithfully represents the vital interests of Israel” — a vivid reminder of his chilly relationship with the leader of Israel’s most important ally.

While in the past Israeli leaders — including Mr. Netanyahu himself during a previous stint as prime minister — have been punished by Israeli voters for mismanaging their relationships with American presidents, analysts were reluctant to attribute too much of his troubles to Mr. Obama, given the complexities of an election that surprised even the experts.

Still, as Martin S. Indyk, a former American ambassador to Israel, put it, “the Israeli public cares about the relationship, and it didn’t help that he mishandled it, and there was a reminder of how badly he mishandled it on the eve of the election.”

Among the intriguing questions, Mr. Indyk said, is whether Mr. Lapid would insist on concessions for joining a coalition with Mr. Netanyahu, like a freeze in settlement construction. While Mr. Lapid’s party has put its emphasis on concerns like jobs and housing, taking a stand on settlements would signal a shift from the right’s agenda.

Almost no one predicts that a new Israeli government will suddenly allow Mr. Obama to rekindle his first-term goal of a peace agreement between the Israelis and Palestinians. Mr. Lapid’s party did not score its victory by pushing to revive long-moribund peace talks. The political climate on both sides remains hostile to such an effort.

Nor, after the frustrations of his first term, does Mr. Obama appear any more likely to invest heavily in Middle East peacemaking. The president scarcely mentions the subject these days.

While Mr. Indyk said that Senator John Kerry, the Massachusetts Democrat who has been nominated to succeed Hillary Rodham Clinton as secretary of state, would make a game effort to preserve the two-state solution, he is no more likely to achieve a breakthrough than Mrs. Clinton did.

Mr. Miller, now at the Woodrow Wilson International Center for Scholars, said he had rarely seen a relationship as persistently dysfunctional as that between Mr. Obama and Mr. Netanyahu. A resounding Netanyahu victory would only have exacerbated those strains.

Now, though, in the wake of his deflating victory, Mr. Netanyahu may have the chance to mend fences, Mr. Miller said.

“The good news for Bibi, if he manages to put it together, is that a broader government would ease tensions and make the next four years much less rocky,” he said. “Netanyahu will be able to preside over a much more functional relationship with the United States.”