Like its predecessors in trying to solve the vexing riddle of Middle East peace, the Obama administration has sought to manage expectations for progress.

But despite the best efforts of Washington, parties on all sides of the conflict now anticipate an early moment of truth for the young administration: the possibility that President Barack Obama, Israeli Prime Minister Benjamin Netanyahu and Palestinian President Mahmoud Abbas will announce an agreement next week, on the sidelines of the U.N. General Assembly meeting, to relaunch Israeli-Palestinian peace negotiations.

With that virtual deadline in mind, and Obama’s Middle East envoy, George Mitchell, on a last-ditch trip to the region to try to secure Netanyahu’s commitment to some sort of settlement freeze, all parties appear to be digging in and trying to bargain as hard as possible, including for the benefit of their domestic political situations.

But interviews with sources from the region and in the U.S. in recent days suggest that the parties, while staking out tough positions in the final days, do not want to see the process collapse or the expected New York “triangle” meeting not be held. They are bargaining hard — playing a game of chicken, as one expert described it — with an eye toward ultimately showing up and with the expectation that relaunched peace negotiations will get under way, numerous sources say.

“The parties — the Israelis, Palestinians and Arab states — are all digging in their positions and doing so very publicly,” says The American Task Force for Palestine’s Hussein Ibish, author of the new book “What’s Wrong With the One State Agenda?” “Either they are all making a big show for domestic public opinion before essentially agreeing to get in line with Obama’s reconvening of the talks ... or they are jockeying for position in a game of chicken.”

In an effort to lower expectations and perhaps engage in its own negotiating gambit, the White House insists that it is not wedded to a peace talks relaunch announcement next week and that such expectations are artificial. “There is no meeting in the books,” a White House official said on condition of anonymity. “The UNGA target [came about] because it was convenient: [The leaders] were all going to be in the same city. It was never a hard deadline.”

Mitchell, who arrived in Israel on Saturday, extended his trip by one day until Wednesday, in part because of a day of mourning in Israel on Monday to mark the death of Israeli fighter pilot Assaf Ramon, the son of Israeli astronaut Col. Ilan Ramon, who was killed in the 2003 Columbia explosion.

Mitchell “is in the final diplomatic stretch,” a U.S. official said. After meeting with Netanyahu for two hours Tuesday, Mitchell went to Ramallah to meet with Abbas. He’s expected to have another meeting with Netanyahu on Wednesday, the official said.

An Israeli official told POLITICO on Tuesday that Netanyahu and Mitchell are likely to compromise on a nine-month freeze of Jewish settlement construction in the West Bank. The official suggested Netanyahu may be arguing about what happens at the end of that period — whether he can go to Bush-era understandings on natural settlement growth, apparently not agreed to by the Obama administration.

Netanyahu has also publicly insisted that East Jerusalem not be considered part of any settlement freeze, although privately, some Washington Middle East hands say, he may have pledged not to embarrass the Americans with more evictions of Palestinians from their homes or provocative new construction in East Jerusalem.

Netanyahu is currently expected to be in New York for the UNGA only on Sept. 23 and 24, the Israeli official said. He said the expectation on the Israeli side is that there would be a trilateral meeting involving Obama, Netanyahu and Abbas at the Waldorf Astoria, where the State Department has a suite on the 27th floor, on one of those days.

Obama will announce that the U.S., with the Israelis and Palestinians, has arrived at a formula to reopen negotiations, the Israeli official suggested. Obama may also announce some general parameters of such negotiations — that they will be based on a two-state solution and adhere to the so-called road map. He may also offer a very general timetable without overly committing himself, the official said.

Under that expectation, negotiations between Israel and Palestinians would get under way sometime soon after that, likely in the region. Several reports suggest the Palestinians are likely to be represented by veteran negotiator Saeb Erekat. Israeli officials say it’s not clear who will be chief negotiator on the Israeli side — suggesting it could be either Netanyahu national security adviser Uzi Arad or Israeli Foreign Minister Avigdor Lieberman. Sub-working groups on specific issues would be led by other designees.

Also being discussed is enlisting Egypt’s Hosni Mubarak to make an announcement around the time Israeli-Palestinian peace talks relaunch. That statement would indicate the commitment for some Arab states to show goodwill and steps toward normalization with Israel for the duration that it commits to freeze settlements and engages in peace talks toward establishing a Palestinian state.