Secretary of State John F. Kerry said Sunday that five months of intensive U.S.-
brokered peace talks have made progress toward resolving the hardest issues dividing Israel and the Palestinians but that a deal could slip through his hands.

“The path is becoming clearer. The puzzle is becoming more defined. And it is becoming much more apparent to everybody what the remaining tough choices are,” Kerry said after three days of shuttle diplomacy in Israel and the West Bank.

Later Sunday, Kerry met with Jordan’s King Abdullah II in Amman and Saudi Arabia’s King Abdullah in Riyadh. The two are key Arab players whose support would be crucial to making a deal stick.

In Riyadh, Kerry praised the Saudi monarch’s long support for a regional peace accord that could end Israel’s many conflicts with Arab neighbors. Abdullah proposed a comprehensive Arab peace accord in 2002 that Kerry said is “part of the framework we have been piecing together.”

The 2002 initiative calls for Israel to give up land taken in the 1967 war, and Israel has never accepted that as the basis for negotiations. In an important amendment last year, the proposals’ backers buttressed Kerry’s peace effort by saying that the region’s 1967 lines could be adjusted by mutual agreement.

The United States is seeking agreement on an outline for a final peace deal that Kerry said he wants to forge by the end of April. He has made 10 trips to the region to push both sides to compromise on borders and other divisive issues that have calcified over decades of conflict.

“I cannot tell you when, particularly, the last pieces may decide to fall into place or may fall on the floor and leave the puzzle unfinished,” Kerry said in Jerusalem.

His idea for a framework agreement on which to build the final peace deal is a tough request for Israeli Prime Minister Benjamin Netanyahu and Palestinian Authority President Mahmoud Abbas. Both sides have balked at Kerry’s terms, according to Israeli and Palestinian news reports. ­Netanyahu and Palestinian chief negotiator Saeb Erekat each used appearances with Kerry over the past few days to accuse the other side of being the potential spoiler in the bid for a deal.

Direct Israeli-Palestinian negotiations resumed in July after a three-year hiatus. The last substantive talks had broken down two years earlier. Kerry has devoted much of his first year as secretary of state to resuming talks and keeping them going. His main claim of progress is that discussions have not ceased.

“This has been a productive couple of days with very, very intensive talks,” Kerry told reporters Sunday. He called the latest talks positive but acknowledged that the effort is at a difficult juncture.

“These are complicated issues that involve the survival and the future of peoples. And this is a conflict that has gone on for too long, so positions are hardened. Mistrust obviously exists at a very high level, so you have to work through that and around that,” Kerry said.

Netanyahu and Abbas have yet to meet face to face, and U.S. officials have said the men are unlikely to do so until their negotiators agree on a framework plan.

“Now is not the time to get trapped in the sort of up and down of the day-to-day challenges,” Kerry said. “This does not lend itself to a daily tick-tock. We don’t have the luxury of dwelling on the obstacles that we all know could distract us from our goal. What we need to do is lift our sights and look ahead and keep in mind the vision of what can come, and if we can move forward.”

Kerry’s call to avoid public criticism and the daily trading of barbs and threats was immediately ignored by Israeli politicians.

On Sunday morning, Yuval Stein­itz, Israel’s intelligence minister and a close ally of Netanyahu, told Israel Radio that Israel would not accept any peace deal based on the pre-1967 lines — a reference to the Green Line, a demarcation established after Israel’s independence that marks the boundary between Israel and the Palestinian territories.

Using the pre-1967 lines with mutually agreed-upon land swaps between Israelis and Palestinians has been a core proposal for peace from the Obama administration.

Ayelet Shaked, a member of the Knesset from the Jewish Home party who is part of Netanyahu’s coalition government, said Sunday, “An Israeli government that would agree to revert the national border to those of 1967 would be performing national suicide.”

Possible borders for a future Palestinian state was not the only issue drawing fire within a few hours of Kerry’s departure. Israeli Foreign Minister Avigdor Lieberman told an annual gathering of Israeli diplomats that a future Palestinian state will have to absorb “hundreds of thousands of Palestinian refugees from Syria and Lebanon because these states will simply expel all of these refugees.”

He also said, “I will not support any peace deal that will allow the return of even one Palestinian refugee to Israel.” His remarks were distributed by his office.

Lieberman repeated one of his past proposals to give to any future Palestinian state a triangle of land in northern Israel that is populated mostly by Arab-Israelis, who make up about 20 percent of Israel’s citizens. In the triangle, he said, the Arab population would not be evicted — but the border would be redrawn and they would end up in Palestine.

Lieberman called the ideas contained in Kerry’s framework agreement “clear and decisive” and said they are probably the most favorable terms Israel will see. Every alternative offered by the international community for a future peace deal will be tougher for Israel, Lieberman said.