Kushner, in his first interview with a Palestinian newspaper, said Washington would be willing to engage with Abbas if he returns to negotiations with Israel, but if he is not willing, the United States will likely announce its peace plan with or without the PA president.
Kushner's direct appeal to the Palestinians - through an interview with the Palestinian Al-Quds newspaper published on Sunday - drew sharp criticism from several senior Palestinian officials.
The interview came out after a trip to the region by Mr Kushner and U.S. envoy Jason Greenblatt. The team met with leaders of Israel, Jordan, Qatar, Egypt and Saudi Arabia to discuss the worsening humanitarian situation in Gaza and the administration's proposals for a peace deal.
Palestinian President Mahmoud Abbas' spokesman Nabil Abu Rudeinah said on Saturday that the US Middle East plans that ignore their rights and the establishment of the Palestinian state will lead to a "dead end". The Palestinians want their own future state with a capital in East Jerusalem.
In response, Kushner said that "Palestinian leadership is saying those things because they are scared we will release our peace plan and the Palestinian people will actually like it". He has 25 years says the same thing.
Saeb Erekat, the Palestinians' chief negotiator, said: "This (U.S.) administration is really trying to destroy the Palestinian moderate camp".
"The way in which you opened your doors to hundreds of thousands of refugees from Syria, not to mention your longstanding commitments to Palestinian refugees, is remarkable", he said.
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Kushner insisted that the peace plan was "almost done", but he offered few details except the vague promise of economic prosperity for the Palestinians.
Israeli Prime Minister Benjamin Netanyahu says his talks with Kushner were productive.
It said, in an official statement, that the Israeli army killed 131, and 14,811 were wounded, including 7,975 treated in hospitals and 54 had their either upper or lower limps amputated.
In an effort to maintain balance during his trip, he will meet Palestinian as well as Israeli leaders.
The U.S. plan is expected to propose solutions to core issues in dispute between the Israelis and Palestinians, such as borders, the future of Israeli settlements, the fate of Palestinian refugees and security.
On Sunday, the IDF revealed evidence demonstrating that the Hamas terror group organizes and carries out kite-based arson attacks originating in the Gaza Strip, refuting claims that the terror activity represents the spontaneous demonstrations of young Gazans.
Sources close to Abbas have apparently revealed that Saudi Arabia and Jordan are among those which have given the green light to talks with the USA administration about the deal, which it is said will end the Israeli-Palestinian conflict.
Major protests in Gaza have seen dozens of Palestinians killed.