Mr. Greenblatt got a very different message at an earlier dinner with two other prominent Palestinians, Salam Fayyad and Ziad Asali, and two American Jewish diplomats, Elliott Abrams and Dennis B. Ross. They all told him that a breakthrough was not realistic now, and that Mr. Trump would be better off pursuing incremental advances, like bettering the economic fortunes of the Palestinians.
“There is a perception that he’s fundamentally sympathetic, but there is an uncertainty about where he wants to go,” said Mr. Ross, a Middle East envoy for Presidents Barack Obama and Bill Clinton. “Among those who think there is no such thing as a deal, or that Israel is being asked to make troubling concessions, there is unease.”
So far, none of these objections are being made public. Conservative supporters of Israel view Mr. Trump as a vast improvement over Mr. Obama, whose blunt pressure on Israel to halt construction of settlements in the West Bank poisoned his relationship with Prime Minister Benjamin Netanyahu.
Mr. Trump has “done more in just 100 days than Barack Obama ever did in transforming the U.S.-Israel relationship into a U.S.-Israel partnership,” said Matthew Brooks, the executive director of the Republican Jewish Coalition.
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But later this month, when Mr. Trump will test his ideas on his first foreign trip, to Saudi Arabia and Israel, one of his most powerful donors, the Las Vegas casino magnate Sheldon G. Adelson, will be in Israel when the president is, according to people briefed on his schedule.
Mr. Adelson was disappointed that Mr. Trump had not fulfilled a campaign promise to move the American Embassy to Jerusalem from Tel Aviv. He will be watching closely to see how the president squares his vow to be a stalwart friend of Israel with his peacemaking ambitions.
The deletion of Mr. Trump’s Twitter message calling it “an honor” to meet with Mr. Abbas was widely noticed by Israeli news media. Curiously, similar messages on other social media were not deleted. Michael Anton, a White House spokesman, said that no one knew what had happened to the Twitter post but that “we stand by the message.”
Mr. Trump has borrowed a few pages from Mr. Obama’s playbook. He, too, has leaned on Mr. Netanyahu to curb settlement construction to make it easier to pursue talks with the Palestinians. Mr. Greenblatt came to an understanding with Mr. Netanyahu that is not unlike the one between President George W. Bush and Prime Minister Ariel Sharon, under which Israel agreed not to approve further construction outside existing settlement boundaries.
After his promise to move the embassy to Jerusalem, Mr. Trump has set that issue aside for now, heeding the advice of King Abdullah II of Jordan and other Arab leaders, who warned him it could ignite violence among Palestinians. White House officials insist it may still happen; Mr. Trump must decide by June whether to renew the waiver of the congressional vote instructing that the embassy be moved.
David M. Friedman, the bankruptcy lawyer who is Mr. Trump’s ambassador to Israel, has told people he plans to divide his time between the ambassador’s seaside residence near Tel Aviv and the King David Hotel in Jerusalem, where the State Department keeps an apartment for its envoy.
But Mr. Friedman, who is moving to Israel in the coming days, has been a less central figure in Mr. Trump’s peacemaking project than Mr. Greenblatt. An Orthodox Jew and an in-house lawyer who negotiated real estate deals for Mr. Trump, Mr. Greenblatt has impressed outsiders with his determination to learn the bedeviling history of Middle East peace.
Mr. Greenblatt has met with a wide range of Arab leaders, ambassadors, and other officials, in a crash course that hard-liners worry will leave him sympathetic to those officials’ arguments, for example, on settlements.
The president’s push on settlements unnerved Mr. Netanyahu, according to officials on both sides, though they avoided an open split over it. Mr. Netanyahu was determined not to antagonize another president, and the understanding on settlements was left unwritten, mitigating attacks on Mr. Netanyahu by pro-settler factions in his coalition.
One of the biggest cheerleaders for a peace deal is Mr. Lauder, a cosmetics heir who has known Mr. Trump for decades. His mother, Estée Lauder, was among the first of the Manhattan social elite to accept Mr. Trump when he arrived on the scene as a young developer from Queens.
Mr. Lauder’s enthusiasm for a bold new initiative has alarmed some in Mr. Trump’s circle, particularly his chief strategist, Stephen K. Bannon. Mr. Bannon speaks regularly with Mr. Adelson, who donated millions of dollars to outside groups to help Republicans in the 2016 elections.
How much direct influence Mr. Lauder has on Mr. Trump is a matter of debate: Some close to the White House insist their contacts are frequent; others say there have been just three meetings, one in the Oval Office and two at Mar-a-Lago, Mr. Trump’s Palm Beach club. But Mr. Lauder has an open line to Mr. Greenblatt; the dinner for Mr. Abbas testifies to his social clout.
An adviser to Mr. Lauder denied he was pushing an alliance with Mr. Abbas, but described Mr. Lauder as an optimist.
Mr. Adelson was frustrated that Mr. Trump did not fulfill his promise to move the embassy on “Day 1.” But people who have spoken to him said he was heartened by the White House’s silence when the Israeli government announced a new settlement to replace Amona, a settler outpost evacuated after it was declared illegal. A spokesman for Mr. Adelson declined to comment.
Aides to Mr. Trump said they are well aware of the hurdles to a deal. But they say the president has an obligation to try, suggesting that his unconventional approach to diplomacy might unlock some doors.
His hard-line pro-Israel supporters console themselves that Mr. Trump will soon recognize the futility of this undertaking. That pessimism, they note, is a view shared not just by hard-liners but also by most of the Israeli political establishment, left and right.
“The administration is likely to discover what its predecessors learned: that there is no deal to be had right now because the parties have unbridgeable positions on most of the issues,” said Noah Pollak, a Republican strategist who works with pro-Israel conservative groups.
“Obama used the impasse as a way to condemn Israel,” he continued. “We’re not worried Trump will follow suit. We simply hope the process of the administration proving to itself that no deal is possible will be quick and undramatic.”
