WASHINGTON — Hamas and the threat it poses to Israel, relegated to the back burner this summer by more pressing challenges from Hezbollah and Iran, are bubbling back into Western policy considerations.

The fate of the Palestinian Authority and the dilemma of how to circumvent the terrorists that lead it — while funneling funds to a Palestinian society on the verge of collapse — were high on the Israeli and U.S. agendas as world leaders convened in New York to address the General Assembly during its opening session.

European enthusiasm for Palestinian efforts to forge a national unity government and a U.S. Congress mired in a campaign season signal cracks in the wall of support Israel has enjoyed for the policy of cutting off Hamas.

President Bush included his vision for the Palestinians in his speech to the General Assembly. Mahmoud Abbas, the relatively moderate P.A. president, was also in New York touting a possible national unity government with Hamas and seeking international aid.

In his speech, Bush said he directed his secretary of state, Condoleezza Rice, to “lead a diplomatic effort to engage moderate leaders across the region, help the Palestinians reform their security services, and support Israeli and Palestinian leaders in their efforts to come together to resolve their differences.”

British Prime Minister Tony Blair “has indicated that his country will work with partners in Europe to help strengthen the governing institutions of the Palestinian administration,” Bush said. “Countries like Saudi Arabia, Jordan and Egypt have made clear they are willing to contribute the diplomatic and financial assistance necessary to help these efforts succeed.”

Bush’s proposals were considered by the Quartet, which released a statement on the issue.

The statement welcomed the Palestinian efforts to form a national-unity government; called on Israel to turn over more than $500 million in tax and customs revenue; and agreed to extend and expand funding to the Palestinians, while continuing to bypass the Hamas-led Palestinian Authority.

JTA has learned that administration officials made it clear to Israel’s foreign minister, Tzipi Livni, that consideration of the Palestinian issue was inevitable at the United Nations this week, no matter how pressing Israel considers the threat from Iran and Hezbollah.

Livni met recently with Bush and other top officials in Washington. She emerged from her talks with Rice with a dual message: Israel wants to help Abbas raise his profile, but not at the expense of funding Hamas, as long as the group continues to reject Israel’s existence and embraces terrorism.

“This is a moment in time in which Mahmoud Abbas has to decide whether the Palestinian Authority will operate on his terms or on the terrorists’ terms,” Livni said. “This is the decision that he has to make. And this is the decision that any future Palestinian government has to make.”

Livni and Abbas met, but details of their meeting were not available.

Livni repeatedly stressed that the problem is not Hamas per se but its platform. Bush administration officials echoed that line.

“We support voices of moderation,” Stephen Hadley, Bush’s national security adviser, said in a pre-General Assembly briefing with reporters. “Obviously, President Abbas is one. He’s committed to peace, and we have worked with him and would continue to work with him. That’s why the president is going to see him. The big question, of course, is whether Hamas will renounce violence, accept the existence of Israel, and accept the agreements that have been made. That’s the $64 question.”

After their meeting in New York, Bush praised Palestinian Abbas for his attempts to reduce the Israeli-Palestinian conflict. “I can’t thank you enough for the courage you have shown,” Bush said.

For his part, Abbas used the opportunity to emphasize his need for aid.

“We look forward to your support” because “we are in dire need of your help and support,” he said with Bush by his side. The two men did not publicly address the issue of a possible Palestinian unity government.

Another question is whether Hamas will really need to change at all. European leaders say they’re steadfast in denying the Palestinian Authority money until Hamas accepts Israel’s right to exist and renounces terrorism, but their eager embrace of Abbas’ plans for national unity suggests that their resolve could be weakening.

A national unity government “creates a new situation,” said Erkki Tuomioja, the foreign minister of Finland, which currently holds the E.U. presidency. “It allows us to break the deadlock.”

The Europeans are increasingly concerned by reports of a Gaza on the verge of anarchy, the result of nonpayment of salaries by the P.A. government, by far the largest employer in the Palestinian areas.

Tough measures in Congress that would severely limit U.S. assistance to the Palestinian Authority and to nongovernmental organizations that help the Palestinians are languishing because of differences between a U.S. Senate bill and its counterpart in the House of Representatives. The Senate version has an exemption for Abbas and allows Bush greater flexibility in bypassing the ban on funds.

“I see a discussion taking place as soon as the new Congress is formed” in January, Sen. Frank Lautenberg, D-N.J., who is Jewish, said in a conference call with Jewish media. “I don’t see there’s enough time to resolve the differences.”

The Conference of Presidents of Major American Jewish Organizations is attempting to fill that vacuum, using the General Assembly to urge Quartet members — particularly the European Union — to closely examine the conditions for any Palestinian unity government.

“We are deeply concerned by reports that the European Union is reconsidering its commitment to the Quartet’s conditions,” the Presidents Conference said in a statement. “Appeasement will only vindicate Hamas’ intransigence and signal to the Palestinians, and to the wider Arab world, that they can reject Israel, continue the terror and still receive European funds.”

The European Jewish Congress is also pressing the case. In a meeting with Finnish Prime Minister Matti Vanhanen on Tuesday, EJC President Pierre Besnainou told Vanhanen that if a new Palestinian national unity government is formed, it must also secure the release of the abducted Israeli soldier Cpl. Gilad Shalit.

“Abbas’ leadership must be strengthened,” Besnainou said in a statement, recalling his recent meeting with Abbas, when the Palestinian leader said Shalit is in good health.

The European Union quickly latched on to news of a possible unity government, which Hamas took as confirmation that the Europeans — who used to be the Palestinian Authority’s main funder before Hamas took power — would resume funding.

But E.U. officials said they would need three months to “reflect” on a new P.A. government’s positions before determining whether to resume aid to the Palestinian Authority, much of which has been rechanneled to humanitarian organizations.

Since July, the European Union has been making emergency aid payments to the Palestinians, including $114.5 million for hospital supplies, fuel and salaries to 625,000 Palestinians whom the government can’t afford to pay.

Abbas’ office gets $15.3 million from the European Union, and European governments provide direct aid adding up to $98 million.

“If you think there is tension between the U.S. and the European Union on resuming funding, you are quite wrong,” said a German diplomat who has been present at U.S.-E.U. discussions and who spoke on background. “In this case there might be nuances, but the E.U. and the U.S. will speak with one voice through the Quartet and their demands will be the same, even if the language sounds different.”

Yet crucial differences could emerge over what constitutes recognition of Israel, a key requirement for resumption of funding. In the past, Palestinians have sought vague formulas that stop short of recognizing Israel, while outside observers have sought to divine in them some unstated acknowledgment of Israel’s right to exist.

“I think the E.U. is looking for the implicit, not just the explicit. And if the unity government stands by all peace agreements, that’s an implied recognition of Israel,” said one source familiar with E.U. declarations on the Middle East.

The United States has been firmer in insisting that if the Palestinians really do accept Israel, they must state it plainly and not obfuscate with formulas that can mean different things to different people.

JTA Correspondent Dinah Spritzer in Prague contributed to this report.