Ahmed Majdalani, a senior Palestinian Liberation Organisation official who is close to Abbas, told AFP on Thursday that one option if no progress is reached would be to dissolve the Palestinian Authority - a threat that has been made in the past.
The report of the planned USA proposal came amid recent signs of disillusionment from Abbas over U.S. peace efforts.
Trump is "very committed to achieving a solution here that will be able to bring prosperity and peace to all people in the area", Kushner said ahead of his meeting with Netanyahu, which took place earlier Thursdayy.
However, the White House is yet to lay out a framework for negotiations following multiple meetings with Israeli and Palestinian leaders.
In addition to Kushner and Abbas, the Ramallah meeting was attended by special envoy for worldwide negotiations Jason Greenblatt and Deputy National Security Adviser Dina Powell, Secretary of the Executive Committee of the Palestine Liberation Organization Dr. Saeb Erekat, Presidential Advisor for Economic Affairs Mohammed Mustafa, head of the General Intelligence Service Maj.
Despite his recent pessimism, Abbas sounded a positive note during his meeting with Kushner.
Both sides and the mainstream media tried to give a positive spin, but there's every indication that last week's meeting between USA special envoy Jared Kushner and Palestinian leader Mahmoud Abbas was just as stormy as the first time the two men got together back in June. "We know that things are hard and complicated، but there is nothing impossible with good efforts،" he added.
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"We affirm that this [US] delegation is working toward peace, and we are working with it to achieve soon what Trump called the 'peace deal, '" he added.
Hamas has blasted Abbas for his pressure moves, accusing him of besieging the Strip alongside Israel.
In past weeks, skepticism was sounded by Palestinian officials about US peace efforts, but both Netanyahu and Abbas have expressed willingness to cooperate with American efforts to renew the talks. "We hope they can work together, live together for many years and have a much better life".
The visit comes with both Palestinian president Mahmud Abbas and Israeli Prime Minister Benjamin Netanyahu not in position to make major concessions, some analysts say, and no details have emerged of how Trump's team would overcome that.
"I have met with Trump envoys about 20 times since the beginning of his term as president of the United States", Abbas said in a meeting with the left-wing Meretz Party in Israel, according to a report in Haaretz.
Netanyahu for his part told Kushner he believed peace was "within our reach".