In an interview with the Wall Street Journal published on Thursday, US President George Bush said he would not deal with Hamas unless it abandoned its tough stance on Israel.
"A political party, in order to be viable, is one that professes peace," he said.
The US has encouraged the parliamentary elections as part of its drive to promote democracy and has reiterated that it will accept the results as the will of the people.
However, Hamas is considered a terrorist organisation by Washington because of its policy seeking the destruction of Israel.
Exit polls from Wednesday's voting predicted that Abbas's ruling Fatah faction would beat Hamas into second place but not secure enough support to win an overall majority.
One poll suggested that Hamas had won as much as 40% of the vote, with Fatah winning 46%.
Such an outcome would increase the pressure to have a Hamas presence in government, a prospect that the United States warned, as recently as last week, could affect Washington-backed efforts for an independent Palestinian State.
Sean McCormack, the US State Department spokesman, said: "In terms of who is seated in the Palestinian legislative council, that will be based on these elections." But he told reporters the choice of cabinet is a matter for Palestinian officials.
Hamas refuses to recognise Israel's right to exist and is responsible for many attacks on the country in recent years.
"A political party, in order to be viable, is one that professes peace"
George Bush, the US president
The movement gained support among Palestinians due to the deadlocked peace process, political divisions within Fatah and because it appeared untainted by corruption.
McCormack said the US would continue to work with the Palestinian Authority but would wait and see the nature of the new cabinet and the policies it advocates.
He said there were no plans to remove Hamas from the State Department terror list.