Abbas, weakened by Hamas's control of the Gaza Strip, was expected to seek increased US pressure on Israel to stop settlement building and ease military checkpoints in the West Bank.
Abbas thanked the US president for reviving stalled negotiations with Israel.
"We believe that you actually are truly seeking a true, genuine
and lasting peace in the Middle East," he said, but warned that "we are in a race against time".
On Thursday the US denied a report in the Washington Post newspaper that it had given Israel permission to expand settlements that it would retain as part of a final peace deal with the Palestinians.
Ehud Olmert, the Israeli prime minister, was quoted as saying by the paper that Bush had given a letter to Ariel Sharon, Olmert's predecessor, four years ago, allowing Israel to expand the settlements.
Ghaith al-Omari, the director of the American Task Force on Palestine, told Al Jazeera he believed the US leader did want to help build a Palestinian state.
"He is sincere - the question is - is he able?"
Al-Omari said the US had realised that while Israeli settlements were being built inside the West Bank, there could be no Palestinian state.
Speaking to Arab-Americans in Washington on Wednesday, Abbas said he was committed to a peace deal by the end of the year but that disagreements with Israel remained.
He urged Israel to stop building settlements, saying the issue was the chief obstacle to peace.