Jibril Rajoub, a high-ranking member of Abbas's Fatah party seen as a possible successor to the 81-year-old president, said he believed Trump could be less beholden to certain interest groups than previous US presidents.
He said that in Trump's phone call to Abbas on March 10 - their first conversation since the US election - he told the Palestinian president he wanted to "end the suffering".
Rajoub told journalists: "Even before the phone call, from the very first moment, our president, I think all of us, said, 'OK, Trump was elected by America... and we are ready to co-operate and we are ready to deal and we are ready to be engaged with him."
Solution to conflict
Trump has sent mixed signals over how he will address the decades-old Israeli-Palestinian conflict, which he says he wants to resolve.
He cast uncertainty over years of international efforts to foster a two-state solution to the conflict when he met Israeli Prime Minister Benjamin Netanyahu at the White House last month.
At that meeting, Trump broke with decades of US policy by saying he was not bound to a two-state solution to the conflict and would be open to one state if it meant peace.
But one of his top advisers, Jason Greenblatt, visited Israel and the Palestinian territories last week, holding a range of meetings on both sides, including with Abbas and Netanyahu.
Rajoub said of Greenblatt's visit that "they are in a stage of exploration" and that the Palestinians were waiting to see what Trump's next steps will be.