"It seems Fatah has lost in the districts," one official told Reuters on condition of anonymity.
Half the seats in Wednesday's vote were chosen on a national list and the other half by districts.
A Hamas majority would put the party in position to shape a new Palestinian government - a situation that could dim prospects for restarting peace talks with Israel.
Other reports have also quoted unnamed Fatah and election officials confirming claims from Hamas to have won over 50% of the vote.
Earlier exit polls that had given Fatah a slight lead.
The Palestinian electoral commission has made no official comment saying only that final results will be announced at 1900 on Thursday (1700 GMT).
Palestinian leader Mahmoud Abbas has said he would step down if he could no longer pursue his peace agenda with Israel.
Earlier leading Hamas candidate Ismail Haniyah claimed victory for Hamas when he said: "Hamas has won more than 70 seats in Gaza and the West Bank, which gives it more than 50% of the vote."
He did not clarify how he obtained the result, or how many votes had been counted.
Early exit polls suggested Hamas had gained 40% of the vote to Fatah's 46%.
According to the Palestinian Central Elections Commission voter turnout in Wednesday's election was 77.6%.
Claiming victory, Haniyah said determining the group's future role would now be a priority.
Results have been postponed
until the end of the day
"This issue is of great importance for Hamas and for most of the Palestinian people because the government will supervise the administration of Palestinian political and internal affairs," he said.
Haniyah said the United States had no right to interfere in Palestinian affairs by putting pressure on Abbas not to include Hamas in a new cabinet.
"The issue of the government is an internal Palestinian affair and no external party has the right to intervene with it," Haniyah said.
Israel and the US have rejected dialogue with Hamas, a group which has carried out nearly 60 bombings since the start of a Palestinian uprising in 2000 against the Israeli occupation.
In recent days, Hamas has raised the possibility of indirect negotiations with Israel, and it omitted from its election manifesto its long-standing call to destroy Israel.
But Haniyah rejected international pressure to disarm after the election, saying Hamas will keep its weapons until Israel leaves Palestinian territory.
Hamas has gained popularity among Palestinians not only for its armed resistance to the Israeli occupation but for its charity network and corruption-free image.
Margin of error
Pollsters had earlier cautioned that there could be a large margin of error in their projections.
One poll, prepared by Khalil Shikaki, head of the Palestinian Centre for Policy and Survey Research in Ram Allah, showed Fatah and Hamas would win 58 and 53 seats respectively out of the 132 contested seats.
Gunfire could be heard in Gaza as
Fatah supporters celebrated
Another by the Bir Zeit University gave Fatah 63 seats and Hamas 58.
With widely varying forecasts, both parties had said they were confident of victory.
In several Palestinian towns Fatah supporters took to the streets to celebrate what they believed would be a victory for their side, honking car horns, shouting slogans and in many cases firing guns into the air.
Voting had continued for a further two hours in East Jerusalem after election organisers said the Israeli authorities had prevented some voters from casting their ballot on time.
Hamas said that the extension was specifically aimed at allowing Fatah supporters to vote.
However, that charge was rejected by Ammar Dweik, head of the Palestinian Central Elections Committee, who described the polls as "a feast of democracy", fair, free and transparent.
Khalid Amayreh contributed to this article.