The article was published after the White House confirmed that Kushner and Jason Greenblatt, whom Trump has named as a special representative for global negotiations, met on Saturday evening in Jerusalem with Prime Minister Benjamin Netanyahu of Israel and US Ambassador David Friedman "to continue their discussions that began on Friday".
Kushner visited Jordan, Saudi Arabia, Qatar and Egypt before talks on Friday and Saturday with Israeli Prime Minister Benjamin Netanyahu.
A spokesman for Mr Abbas said U.S. efforts would yield no result if they bypassed Palestinian rule.
"I don't want to launch an operation, but there is a good chance that we will have no other option but to go in so that we can create durable deterrence", he said on Army Radio, also adding that the attackers "should be treated like a terrorist and shot regardless of age".
"The global community is getting frustrated with Palestinian leadership and not seeing many actions that are constructive toward achieving peace", Kushner said. "There is a good deal to be done here from what I assess", Kushner said.
Kushner emphasized the economic benefits of a deal, saying that he believed "we can attract large investments in the infrastructure from both the private and the public sectors" to improve the Palestinian economy.More news: Mo Salah considering quitting Egyptian national team
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It also said Gazans wounded by Israel are being transferred to hospitals in Cairo.
US President Donald Trump's envoys at the weekend tried to bypass the Palestinian Authority by appealing directly to the Palestinian Arab street not to reject a peace proposal that their leaders hadn't even yet examined.
USA -brokered peace talks collapsed in 2014. Palestinians seek east Jerusalem as the capital of a future state.
This Hamas-provoked tit-for-tat resembles similar sequences of events that took place two weeks ago in a violent escalation between the Palestinian Islamic Jihad (PIJ) and Israel.
"The Arab countries made it clear that their position remains clear, namely that a two-state solution should be based on global resolutions [pertaining to the Israeli-Arab conflict]", he added. He has the same talking points that haven't changed in the past 25 years.
While the details of the so-called deal of the century have not officially been released, leaks have suggested that the Palestinians would initially control the Gaza Strip and less than half of the occupied West Bank, while a Palestinian capital would be created from villages surrounding Jerusalem.
Israeli forces and settlers pulled out of the Gaza Strip, now controlled by Abbas's main rival, the Islamist Hamas group, in 2005.
Prince William flies to Jordan Sunday on a trip that will see him become the first member of Britain's royal family to pay an official visit to both Israel and the Palestinian territories.