Palestinian and Israeli negotiators sat down Monday to address their toughest disputes, honoring promises made to President Bush during his visit last week. But talks were undermined by a hawkish Israeli lawmaker's threat to bolt Prime Minister Ehud Olmert's coalition government as he tries to clinch a peace deal. Israeli Foreign Minister Tzipi Livni, Israel's lead negotiator, and chief Palestinian negotiator Ahmed Qureia met Monday for two hours at a Jerusalem hotel, and Israeli officials said such meetings were expected to continue on a more-or-less weekly basis.

"They started talking about the core issues," Israeli Foreign Ministry spokesman Arye Mekel said, using a phrase which refers to the key questions such as the borders of a Palestinian state, the status of Jerusalem and the future of Palestinian refugees.

"It was an exploratory session, and we exchanged our views on how to approach the core issues," Qureia said afterward.

Lawmaker Avigdor Lieberman, head of the hard-line Yisrael Beiteinu Party, threatened earlier this month to quit Olmert's five-party ruling alliance when peace talks moved to those questions, which lie at the heart of the Israeli-Palestinian conflict and trigger deep emotions on both sides.

Lieberman was due to meet Olmert on Tuesday to sound out the prime minister on his intentions. After the meeting, he and party colleagues would decide whether to remain in the government, Lieberman spokeswoman Irena Etinger said.

After a U.S.-sponsored Mideast conference in November, Israelis and Palestinians relaunched negotiations after seven years of violence. They pledged to try to reach an agreement before Bush leaves office a year from now.

The talks stalled immediately over Israeli construction in disputed territory and Palestinian attacks on Israel. But on the eve of Bush's visit, which he said was meant to nudge the sides closer together, Palestinian President Mahmoud Abbas and Olmert instructed their teams to start discussing the thorniest issues.

The new impetus could, however, deepen divisions within Olmert's government, which seats his centrist Kadima party alongside Lieberman's hawks, Labor doves, the hard-line ultra-Orthodox Jewish Shas and a party representing senior citizens in a broad but fragile coalition.

Olmert's government would still command 67 of parliament's 120 seats even if Yisrael Beiteinu's 11 lawmakers were to leave. But Shas, which has 12 seats, has also threatened to jump ship if Jerusalem comes up in negotiations. Shas rejects any compromise over Jerusalem, while Palestinians claim the eastern section as the capital of the state they want to create.

That would leave Olmert scrambling to rebuild a parliamentary majority, most probably by wooing another ultra-Orthodox party and the dovish, passionately secular Meretz party.

Speaking to parliament Monday, Livni said that conducting peace talks with the Palestinians now would give Israel more control of the outcome than later, when an exasperated international community might apply pressure.

"It is true that I am ready to make significant territorial concessions," she told the chamber. "I prefer to get there while I manage the negotiations, not when they put a plan on the table and I have to decide whether to take it or leave it, for better or worse."

Israel is talking peace with Abbas, who heads the Fatah faction, while fighting armed Islamic groups who fire rockets and mortars at southern Israel almost daily from Gaza, which is run by the rival group Hamas.

Even before Livni and Qureia sat down for their talks, Hamas spokesman Sami Abu Zuhri demanded that Abbas "stop giving away Palestinian blood and rights for free."

"This is a failed meeting that is going to provide a cover for the occupation crimes against our people," Abu Zuhri told reporters.

Hamas seized Gaza from Abbas-allied forces in June. The militant Islamic group is isolated internationally and is not a party to the negotiations. Olmert has said repeatedly that Israel would not implement any peace agreement before Gaza militants were subdued.

In remarks Monday to parliament's defense and foreign affairs committee, Olmert reiterated his doubts about Abbas' ability to regain control of Gaza, a participant said on condition of anonymity because the session was closed.

"I'm not sure the man leading the Palestinian Authority will have the power to bring about the implementation of the agreement," Olmert was quoted as saying.

Written By STEVE WEIZMAN