There are growing indications the Islamic militant group Hamas might be heading for a shock victory in the Palestinian elections.
Both Fatah and Hamas have claimed victory in the election
Officials from the ruling Fatah party said Hamas appeared to have won a majority of seats in the legislature.
Exit polls indicate a close race between the two parties. Final results are expected at 1900 (1700 GMT).
Israel, the US and EU consider Hamas a terrorist organisation and have said they do not want to deal with it.
The BBC's Jon Leyne in Jerusalem says a victory for Hamas would present a huge challenge for Israel and the international community and for that matter, for Hamas itself.
He says many believe the militant movement would be more comfortable as the major opposition party rather than having power thrust on it at the first attempt.
With counting still under way, a senior Fatah official told the BBC his party has lost and Hamas has won a majority in the legislature.
The Associated Press news agency quoted unnamed Palestinian election officials as saying Hamas had won virtually all the seats in electoral districts in the West Bank and Gaza.
The districts account for half the 132 seats in the legislature, with the other half chosen from a national list of candidates.
"Hamas has won more than 70 seats in Gaza and the West Bank, which gives it more than 50% of the vote," Ismail Haniya, a leading Hamas candidate, said, according to the Reuters news agency.
Hamas is contesting its first parliamentary election, having boycotted the previous poll in 1996.
The party does not recognise Israel and has launched hundreds of attacks against its citizens.
Three exit polls put Fatah ahead by a narrow margin, but correspondents say whatever the result, Fatah is no longer the single dominant force in Palestinian politics.
Fatah - founded by the late Palestinian leader Yasser Arafat - has controlled the 132-seat Palestinian parliament since its inception.
Palestinian leader and Fatah chief Mahmoud Abbas has said he is ready to start peace talks with Israel even if Hamas joins the government.
"We are partners with the Israelis. They don't have the right to choose their partner. But if they are seeking a Palestinian partner, this partner exists," he said.
But Israeli acting Prime Minister Ehud Olmert, quoted by AFP news agency, said before exit polls were released that Israel could not allow Hamas in its current form to become part of the Palestinian Authority.
US President George W Bush also said Washington would not deal with members of Hamas, even if they ended up in positions of responsibility, unless the group renounced a desire to destroy Israel.
"A political party, in order to be viable, is one that professes peace, in my judgment," he said in an interview for the Wall Street Journal.
White House spokesman Scott McClellan said: "We do not deal with Hamas. Hamas is a terrorist organisation. Under current circumstances, I don't see any change in that."
The BBC's Justin Webb in Washington says that while the White House is not threatening to break off relations with the Palestinians if Hamas joins the cabinet, it is making it very clear it would prefer the group to be kept in opposition for the time being.