The White House talks came with the Arab world still livid about the Israeli raid on the aid flotilla on May 31, which killed nine activists, and amid frantic US efforts to stop the uproar derailing peace efforts.
"I believe that potentially we can make some significant progress before the end of the year," Obama said, telling Abbas in the Oval Office he remained deeply committed to spending personal political capital in the Middle East.
The US leader said that the 400-million-dollar aid package was meant to reaffirm US commitment to improving the "day-to-day lives of Palestinians."
He said it may be possible to take the "tragedy" over the Gaza aid convoy and turn it "into an opportunity to create a situation where lives in Gaza are actually, directly improved."
Abbas said that the aid package was "positive."
But he told Obama in front of reporters that he needed to see progress in the current US-mediated "proximity" talks with Israel, before proceeding to direct discussions.
"We are working in order to make progress. President Obama is working for that, to see progress," Abbas said.
As Obama seeks to inject new momentum into peace moves, both the White House and the Israeli government said they were trying to finalize arrangements for a visit to Washington by Israeli Prime Minister Benjamin Netanyahu this month.
The Israeli leader had been due here last week, to smooth over tensions which flared during his last visit, but had to cancel because of the Gaza crisis.
Chief Palestinian negotiator Saeb Erakat, who was in the Oval Office meeting, meanwhile told AFP that US envoy George Mitchell would go back to the Middle East next week for more proximity talks.
Erakat said Abbas had come to the United States with the message that progress was urgently needed.
"Time is of the essence, that's his message, we need to see genuine movement in the direction of a two-state solution and ending the occupation," he said.
The US leader also called on Israel to live up to UN Security Council conditions on probing the flotilla raid, which laid out the need for "credible, international involvement."
"I've said to the Israelis directly ... it is in Israel's interest to make sure that everybody knows exactly how this happened so that we don't see these kinds of events occurring again," Obama said.
But Israel has rejected any international inquiry into the affair, amid calls for an easing of the three-year blockade of Gaza.
Earlier, Netanyahu said he was in talks with "several members of the international community" but said a probe should not focus on the role of Israeli soldiers in the raid.
Obama, while saying Israel had a right to make sure arms were not infiltrated into Gaza, also said that it should be possible to work out some kind of framework to avoid a repeat of the flotilla raid.
"It seems to us that there should be ways of focusing narrowly on arms shipments, rather than focusing in a blanket way on stopping everything and then, in a piecemeal way, allowing things into Gaza," he said.
The White House said the aid package for Palestinians would help "increase access to clean drinking water, create jobs, build schools, expand the availability of affordable housing, and address critical health and infrastructure needs."
The money will be largely be funneled through the US Agency for International Development and the United Nations Relief Works Agency, which deals with Palestinian refugees.
The United States avoids dealing with Gaza's ruling Hamas, which it deems a terrorist organization, and has worked to shore up the West Bank government of Prime Minister Salam Fayyad.