Several officials said this could be the first trip in which Mr. Kushner and Mr. Greenblatt delved into the nitty-gritty of a possible peace agreement — borders, security and other questions that have bedeviled American peacemakers for decades — by asking both sides to list their priorities for negotiations.

“Part of it is to figure out how to make incremental change that results in a lasting peace,” Sean Spicer, the White House press secretary, said on Monday. “Part of this is really to utilize the trust that has been built up, and not have these negotiations out in public. But I think that they had a very successful visit when the president was over there, and they’re going to continue to build on that.”

There have been growing expectations, fueled in part by Mr. Trump’s optimistic pronouncements, that the White House will invite the two sides to Washington in the coming months to negotiate directly. But two officials said that was not the goal of this trip.

Mr. Greenblatt has already made a handful of visits to Israel and the West Bank and obtained an unwritten agreement from the Israeli government to slow down construction of Jewish settlements, which have been an impediment to previous American efforts to broker a deal.

A real estate lawyer who worked for the Trump Organization and had no diplomatic experience, Mr. Greenblatt has nonetheless won praise from both sides for being open-minded. But the test will come now, as he begins to press the Israelis on issues like the contours of a Palestinian state and demands that the Palestinians halt language that incites violence against Jews.