Abbas said his meeting with Trump on the sidelines of the UN General Assembly meeting in New York attested to the US president’s seriousness about reaching a Middle East peace deal soon.
"If this is any proof to anything ... it attests to the seriousness of your excellency, Mr. President, to achieve the deal of the century in the Middle East during this year or in the coming months, God willing," Abbas said through an interpreter.
Trump, meanwhile, pledged to devote everything "within (his) heart and soul" to bringing peace to the Middle East.
"We're looking seriously at peace and maybe ultimately peace in the whole of the Middle East. I think we have a pretty good shot—maybe the best shot ever—and that's what we're looking to do," he said ahead of his meeting with Abbas.
"It's a complex subject," the American president admitted. "Always been considered the toughest deal of all... But I think we have a very, very good chance, and I certainly will devote everything within my heart and within my soul to get that deal made."
Trump said his team of advisers was working very hard on the issue, as were Israel, Saudi Arabia and other nations.
"So we'll see if we can put it together. Who knows? Stranger things have happened, but I think we have a good chance," he asserted.
"We're at a very important juncture, there's a small period of time, and we're going to see what we can do. There can be no promises, obviously," Trump added.
Abbas noted that the Palestinians had met with Trump's team more than 20 times since the US president entered office in January, and he said the fact that Jews and Muslims were celebrating a new year was a positive sign.
"This is a very sweet coincidence that we can celebrate the new year together within a 24-hour period, and if this is an indication to anything, it means that we can coexist peacefully together," Abbas said.
The Palestinian Authority's ambassador to the United States, Husam Zomlot, said "the meeting was forward-looking, and the conversation was deep and frank between the two allies."
The Palestinian Authority said in a statement Abbas told Trump that "peace can be achieved through implementing the two-state solution based on the 1967 borders." Abbas also said "the illegal Israeli settlements policy poses an immediate threat" to that.
Trump neglected to mention the Israeli-Palestinian conflict or the Mideast peace effort, focusing instead on the nuclear threats from Iran and North Korea.
Prime Minister Benjamin Netanyahu did mention the Palestinians, but only briefly, saying "Israel is committed to achieving peace with all our Arab neighbors, including the Palestinians."
UN Secretary-General António Guterres did make note of the Israeli-Palestinian issue, saying during his General Assembly speech Tuesday, "Last month, I visited Israel and Palestine. We must not let today’s stagnation in the peace process lead to tomorrow’s escalation. We must restore the hopes of the people. The two-state solution remains the only way forward. It must be pursued urgently."