Trump spoke after welcoming Abbas to the White House for a meeting aimed at relaunching talks that have failed repeatedly to resolve the decades-old conflict.
"I've always heard that perhaps the toughest deal to make is the deal between the Israelis and the Palestinians," Trump said. "Let's see if we can prove them wrong."
"We will get it done," he said.
Speaking through a translator, Abbas echoed Trump's upbeat tone, saying, "We believe that we can... Be true partners to you to bring about a historic peace under your stewardship."
The long-shot effort to bring peace -- which has eluded US presidents since the 1970s -- got off to a rocky start early in Trump's administration.
Trump renounced support for a Palestinian state and vowed to move the US embassy to Jerusalem, breaking two tenets of American policy held for decades.
Vice President Mike Pence today said Trump is still "giving serious consideration into moving the American embassy in Tel Aviv to Jerusalem."
At the same time, Trump has urged Israel to hold back on settlement building in the West Bank, a longstanding concern of Palestinians and much of the world.
Pence said Trump is "personally committed to resolving the Israeli and Palestinian conflict" and "valuable progress" is being made.
The 82-year-old Abbas made the trip to Washington while politically unpopular back home, with polls suggesting most Palestinians want him to resign.
Abbas's term was meant to expire in 2009, but he has remained in office with no elections held.
Palestinians are watching closely, including the leader of the radical Hamas movement Khaled Meshaal, who called on Trump to break with past approaches to Middle East peace.
"This is a historic opportunity to pressure Israel... To find an equitable solution for the Palestinian people," he told CNN.
Hamas yesterday released a policy document that for the first time indicates a willingness to accept the idea of a Palestinian state within the borders that emerged after the 1967 Arab-Israeli war, although it stops short of recognizing Israel.
One of Trump's top advisers, Jason Greenblatt, held wide-ranging talks with both Israelis and Palestinians during a visit in March.
Abbas and Trump spoke by phone on March 11, and there are suggestions the US president could visit the Middle East this month.
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