"We highly appreciate President Trump's efforts to strike a historical peace deal, a statement he repeated more than one time during our meetings in Washington, Riyadh and Bethlehem", Abbas said at the beginning of the meeting, according to Wafa, the official Palestinian news agency.
In recent weeks Palestinian officials have criticized the administration's unwillingness to publicly back the creation of a Palestinian state or endorse the Palestinians' territorial demands in the West Bank.
After negotiations in Israel, the USA delegation is also scheduled to hold talks with Palestinian President Mahmoud Abbas.
Palestinians were still seeking a pledge of support from the Trump administration for the creation of a Palestinian state alongside Israel - the foundation of US Middle East policy for the past two decades.
White House adviser Jared Kushner on Thursday kicked off a renewed push by the Trump administration to restart Mideast peace talks amid signs of growing Palestinian impatience with a lack of progress so far.
Palestinians, who initially expressed hope that Trump might be able to engineer the peace deal that eluded his predecessors, recently have turned more pessimistic.
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They said: "Every child has the right to safe access to education and States have an obligation to protect, respect and fulfil this right, by ensuring that schools are inviolable safe spaces for children".
Earlier this week, officials in Ramallah told The Media Line that the PA will resume its campaign to gather unilateral indices of statehood recognition at United Nations agencies and other worldwide fora despite the belief shared by the USA and Israel that such unilateral action is harmful to the quest for peace. Extremists on both sides have been trying to discredit the two-state vision since it was internationally adopted. The presidents will discuss the Temple Mount crisis in Jerusalem as well as the peace settlement in Palestine, the news outlet reported, citing Erdogan's spokesman Ibrahim Kalin. "I think that all of them are within our reach".
But at the same time, Majdalani said they could also unilaterally declare Palestinian statehood.
Netanyahu, for his part, is under pressure from his rightwing base not to make concessions to the Palestinians and to continue Jewish settlement building, and there is little incentive at the moment for him to change course, some analysts say.
Shapiro, now a fellow at the Institute for National Security Studies think-tank in Tel Aviv, said the focus should instead be on short-term goals such as improving the Palestinian economy in order to keep the possibility of a two-state solution alive.
"So I am happy to see you and the effort you're leading on behalf of the [US] President [Donald Trump] with Jason [Greenblatt] and other members of your team", the prime minister said.