In a photo op at the start of their meeting, Mr. Netanyahu greeted Mr. Greenblatt warmly, addressing him as Jason, and said, “I hope we can do some good things together.” Mr. Greenblatt replied, “I think we are going to do great things together.” Once seated, he thanked Mr. Netanyahu for reorganizing his schedule, throwing in a “todah,” Hebrew for thank you.
Mr. Greenblatt’s visit followed the first telephone call from Mr. Trump to President Abbas on Friday, in which he invited Mr. Abbas to the White House.
After his meeting with Mr. Netanyahu, Mr. Greenblatt was scheduled to meet Mr. Abbas at his headquarters in the West Bank city of Ramallah on Tuesday.
On the Israeli side, much about Mr. Greenblatt’s visit was confidential. Mr. Netanyahu’s aides did not immediately respond even to basic questions about his schedule.
Israeli officials have confirmed that since Mr. Netanyahu’s visit to Washington last month, Mr. Greenblatt has been trying to formulate understandings on the contours of future Israeli settlement construction in the occupied West Bank. The issue of continued settlement building enrages the Palestinians and became a constant source of tension between the Israeli government and the Obama administration.
Israeli analysts said that Mr. Greenblatt’s talks with Mr. Netanyahu were likely to focus on this more modest issue than on any concrete grand plan to restart Israeli-Palestinian negotiations.
Israel’s strong pro-settler constituency has cheered Mr. Trump’s ascendance, seeing him as an ally. But after Mr. Netanyahu announced thousands of new homes in the West Bank, Mr. Trump asked him to delay new construction, saying it would not help peace efforts.
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Mr. Netanyahu, a conservative, already finds himself in a bind, having pledged to build an entirely new settlement in the West Bank — the first in decades — to appease the settlers after the recent, court-ordered removal of an illegal outpost.
Emboldened by Mr. Trump’s victory, the Israeli right has been asserting itself. On Sunday night, barely a week after the Israeli Parliament passed a contentious law barring foreign activists who call for a boycott of Israel or its settlements, the Israeli authorities prevented Hugh Lanning, chairman of the Britain-based Palestine Solidarity Campaign, from entering the country.
But some Israeli commentators surmise that when it comes to the peace process, the right may be in for an unpleasant surprise. “From a messiah elected by divine miracle to deliver Israel from the injustices perpetrated by his predecessor, Trump could turn out to be the Israeli right wing’s worst nightmare,” Chemi Shalev, a columnist, wrote on Monday in the liberal Haaretz newspaper.
There was some consternation on the Palestinian side when, during Mr. Netanyahu’s visit to Washington, Mr. Trump discarded two decades of American support for the establishment of a Palestinian state alongside Israel, saying he would go along with any form of an agreement that made the Israelis and the Palestinians happy.
But Palestinian officials said that Mr. Trump seemed serious about reaching a peace deal in his conversation with Mr. Abbas and reaffirmed Mr. Abbas’s status and that of his Palestine Liberation Organization as partners for any negotiation.
“We knew that the Israelis tried to exclude the Palestinian side, saying that the leadership does not represent the Palestinian people anymore,” Hanan Ashrawi, a senior P.L.O. official, told the Voice of Palestine Radio on Monday. Mr. Greenblatt’s visit, she said, was “all about exploring the two sides’ positions” in order to find a path back to talks.
For some here, the American envoy’s lack of any diplomatic experience makes his mission all the more intriguing.
Noting that there was a long list of American mediators dating to the 1940s who failed to achieve peace, Michael Oren, a deputy minister for diplomacy in the prime minister’s office, told Israel Radio, “A background of policy-making in this matter is not necessarily a promise for success.”
Mr. Trump seemed to have tapped Mr. Greenblatt as an Israel adviser almost spontaneously when Mr. Greenblatt was called in to help answer questions as Mr. Trump briefed reporters from the Jewish press during the campaign.
A father of six from Teaneck, N.J., Mr. Greenblatt spent a year after high school in the 1980s studying at the Har Etzion Yeshiva in a West Bank settlement south of Jerusalem. In an interview with the Jewish Telegraphic Agency during the campaign, he said he had not met any Palestinians since his stay at the yeshiva, where he had some casual interactions with Palestinian laborers, gardeners and shopkeepers.
He also said that his main sources of information on Israel were daily email alerts, American Israel Public Affairs Committee materials and a weekly Jewish radio program featuring Malcolm Hoenlein, a Jewish communal leader, in addition to conversations with some people involved in the Israeli government. He, his wife and children also contributed to a family travel guide to Israel.
Mr. Greenblatt wrote a post in October on Medium.com berating the Palestinian Authority leadership for showing what he described as “contempt for its Jewish neighbors” after Palestinian security forces detained four Palestinian civilians who had met with settlers during the Jewish Sukkot holiday.
Yet in an appreciation after the death in September of Shimon Peres, Israel’s elder statesman who had become most identified with the pursuit of peace and the two-state solution, Mr. Greenblatt praised Mr. Abbas for his gesture of attending Mr. Peres’s funeral, despite the criticism Mr. Abbas faced from some of his own constituents.
