Brushing aside the complexities of a decades-old conflict that has bedevilled successive USA leaders, Trump told Abbas that together with the Israelis they could bag "the toughest deal to make".
The billionaire said: 'It's. something that I think is frankly maybe not as hard as people have thought over the years.
Trump added that an agreement can not be imposed on the two sides by any foreign country, including the United States, and would have to be negotiated directly between them. "We are the only remaining people in the world that still live under occupation".
The source of Trump's optimism was not immediately apparent. The President was clear that only by working together can the parties reach an agreement but he concluded "we will get this done".
But Abbas will be encouraged that Trump did not publicly demand that the Palestinians cut funding to prisoners and their families.
Asked what distinguishes Trump's plans from previous attempts, White House spokesman Sean Spicer said merely: "The man is different".
Like previous US leaders, Trump faces numerous obstacles in the long-shot bid.
KUTTAB: If it has along with it a freeze of settlement because without a deal the settlements are continuing.
Abbas reasserted the overarching goal of a Palestinian state, saying it must have its capital in East Jerusalem with borders based on pre-1967 lines.
The Palestinians want to set up a state in the West Bank, Gaza Strip and east Jerusalem, lands Israel captured in the 1967 Mideast war.
There have been no serious negotiations since gaps widened with the 2009 election of Benjamin Netanyahu as Israel's prime minister.
But Trump's commitment was met with skepticism in diplomatic quarters, coming just two months after he broke with the longstanding US policy of advocating a two-state solution, and vowed to move the US Embassy to Jerusalem, breaking with US policies that have lasted decades. His comments followed a number of statements from angry right-wing politicians, who were outraged that Abbas could speak of peace while at the same time issuing such payments, which they believe help inspire acts of terrorism against Israelis.
However, at the same time, Mr Trump has urged Israel to hold back on settlement building in the West Bank, which has been a long-standing source of tensions in the region.
"We'll start a process which hopefully will lead to peace", Trump said.
"Some of the topics that were discussed during their meeting and the lunch were: advancing the Israeli-Palestinian peace; preventing incitements to violence, particularly media outlets directly associated with the Palestinian Authority; strengthening efforts to combat terrorism, including defeating Daesh; measures to empower the Palestinian economy and provide economic opportunity for the Palestinian people", he said.