The leaders of Israel and the Palestinian Authority held nearly two hours of face-to-face talks in this Red Sea resort on Tuesday in a second round of negotiations that delved into the core issues dividing them but did not yet break an impasse over Jewish settlements.

President Obama’s special representative to the Middle East, George J. Mitchell, said he was encouraged by the overall direction of the talks but declined to say whether the two sides made any progress on a dispute over Israel’s moratorium on settlement construction, due to expire on Sept. 26.

Prime Minister Benjamin Netanyahu has ruled out extending the moratorium, while the president of the Palestinian Authority, Mahmoud Abbas, has threatened to walk out of the talks if Israel allows it to expire.

But in a sign of the complexity of the talks, Mr. Netanyahu delayed plans to return to Jerusalem after a lunch hosted by the Egyptian president, Hosni Mubarak, to hold an unscheduled meeting with Mr. Abbas, an Israeli official said. The two had already planned to meet again on Wednesday in Jerusalem.

Speaking to reporters after the first meeting, Mr. Mitchell said, “We continue our efforts to make progress, and we believe we are moving in the right direction overall.”

In addition to the Israeli and Palestinian leaders, the meeting included Mr. Mitchell and Secretary of State Hillary Rodham Clinton.

Mr. Mitchell, sticking to his policy of saying little publicly about the negotiations, declined to discuss whether Israel had offered a compromise on the moratorium or whether the Palestinians had signaled any flexibility in their position. But he said the two leaders had held substantive talks on other central topics.

These issues, which have confounded negotiators for decades, include the borders of a new Palestinian state, security guarantees for Israel, the political status of Jerusalem and the rights of Palestinian refugees.

“I’m not going to attempt to identify each one that was discussed, but several were, in a very serious, detailed and extensive discussion,” Mr. Mitchell said. This, by itself, would mark progress: after the first round of talks in Washington, he said there had not been a detailed discussion of these issues.

Mr. Mitchell repeated Mr. Obama’s call on Israel last week to extend the moratorium on settlements “especially,” he said, “given that the talks are moving in a constructive direction.”

“We know that this is a politically sensitive issue in Israel,” Mr. Mitchell said. “We’ve also called on President Abbas to take steps that help encourage and facilitate this process.”

On Monday, Mrs. Clinton suggested that the two sides could find a creative solution to the impasse — steps that would allow the Palestinians to accept less than a full extension of the moratorium or that would enable Mr. Netanyahu to sell an extension to his domestic audience.

“That has to be understood as an effort by both the president and the prime minister to get over a hurdle posed by the expiration of the original moratorium,” Mrs. Clinton said to reporters on the way to Egypt.

The busy day of diplomacy, in a luxury hotel overlooking the sparkling waters of the Red Sea, illustrated the importance of Egypt and other Arab countries. President Mubarak met one-on-one with Mr. Netanyahu and Mr. Abbas. Later on Tuesday, Mrs. Clinton was scheduled to meet with the foreign minister of the United Arab Emirates, Abdullah bin Zayed.

Mrs. Clinton and Mr. Obama have both criticized Arab countries for failing to do enough to support the Palestinian Authority financially, despite their public statements of support for a Palestinian state.

In a sign of the Obama administration’s ambitions for a broader Middle East peace accord, a senior official said Mr. Mitchell would travel later this week to Lebanon and Syria.