Trump's adviser also acknowledged that Arab leaders support a two-state solution, with a Palestinian state existing side-by-side with Israel, but he gave no indication that this was part of Washington's plan.
The visit comes at a particularly volatile time after US President Donald Trump recognized Jerusalem as capital of Israel and moved Washington's embassy there, sparking Arab outrage and deadly clashes.
In Jerusalem, he will stay at the King David hotel, which was Britain's administrative headquarters during its rule of Palestine prior to Israeli statehood in 1948.
Kushner spoke in an interview with the Palestinian newspaper Al Quds, the Arabic name for Jerusalem.
He will hold talks with both Israel's Prime Minister Benjamin Netanyahu and Palestinian president Mahmud Abbas.
Other members of William's family - including his father Prince Charles - have made unofficial visits to Israel and east Jerusalem in the past.
"Catherine's experience is not unique - the interchange between our two countries is real and deep: work, study, tourism and family links".
Although it is privately owned, Al-Quds regularly serves as a mouthpiece for the Palestinian Authority.
"I question President Abbas' ability, or desire, to finish the deal".
Kushner said he is "ready to work with [Palestinian Authority] President Abbas, if he wishes". He has the same talking points that haven't changed in the past 25 years.
"What Kushner did is outrageous and totally unacceptable", a senior Palestinian official in Ramallah told the Post. "The Trump administration thinks it can incite the Palestinians against their leaders".
Kushner appealed directly to the Palestinian people and portrayed Abbas, 82, as a leader entrenched in the past.
Kushner and Greenblatt's trip to Jerusalem followed a regional tour that included Jordan, Egypt and Qatar.
Kushner's interview, Sha'ath said, "was the best proof that the deal of the century has failed".
PLO Secretary-General Saeb Erekat said Kushner's interview "again illustrates the USA refusal to talk substance, to mention Palestinian rights or a Palestinian state".
"This is an attempt to push forward a plan that consolidates Israel's colonial control over Palestinian land and lives while telling the Palestinian people that money will compensate for our inalienable right", he added. Israel will try to project itself as a vibrant, open, high-tech society while the Palestinians will explain what it is like to live under occupation.
Meanwhile, former PA information minister Nabil Amru criticized the PA leadership for cutting its ties with the United States administration.
Al Jazeera's Harry Fawcett, reporting from West Jerusalem, said that the Palestinian leadership's reaction made it clear that there was "no prospect at this stage" that they want "anything to do with the USA administration or its plan, even ahead of its publication". "No one knows why the Palestinian leadership did that because the situation remains unclear until now", Amru told the Palestinian news agency Sama.
"If there's any plan, this it's being implemented on the ground: with moving the U.S. embassy to occupied Jerusalem, withdrawing support for the two-state solution, cutting funds to UNRWA [UN Relief and Works Agency] and, eventually trying to normalize the Israeli apartheid in Palestine", Saeb Erekat said, as cited by Middle East Eye.
"They are working and trying to work hard in a regime change, because Palestinian leadership under the leadership of President Abbas wants genuine, lasting, comprehensive peace, based on worldwide law", Erekat added.