Counting has begun at some polling stations after two days of voting in Egypt's first free elections.
Only a small fraction of results have been announced so far, with counting to continue over the weekend.
At a news briefing, the prominent Muslim Brotherhood claimed that its candidate, Mohammed Mursi, was leading, based on the party's own estimates.
Earlier, tensions were high as some candidates traded accusations, but the poll was praised as largely peaceful.
Around 50m people were eligible to vote in the polls, in which 13 candidates are vying for the presidency more than a year after Hosni Mubarak was toppled from power after 18 days of mass protests.
As well as Mr Mursi, the frontrunners include two former Mubarak-era officials, ex-PM Ahmed Shafiq and former head of the Arab League Amr Moussa, and Abdul Moneim Aboul Fotouh, an independent Islamist candidate.
If no contender gains more than 50% of the vote, the two leading candidates will compete in a run-off next month.
Counting began as soon as polls closed, and some individual polling stations may announce early results. Full results are set to be announced on Tuesday, although correspondents say they are likely to emerge earlier.
Late on Thursday, the Muslim Brotherhood's Freedom and Justice Party told reporters that it was confident that Mr Mursi had gained the most number of votes.
"I am confident that as first indications show, our candidate is leading," Essam al-Erian, vice chairman of the party, said at a news conference in Cairo.
The prediction was based on results from 236 out of 13,000 polling stations, he said.
NGOs and rights groups monitoring the election have reported some complaints, including illegally campaigning outside polling stations.
But overall, electoral observers appeared to be satisfied with the process.
A former United States congresswoman, Jane Harman, who observed the voting, said she was impressed by what she had seen.
"We don't know who will win this election, we don't know whether there will be a run-off, we don't know exactly how the president of this country, what powers he will have and what will happen with the parliament and the constitutional assembly.
"But as one observer of other elections too, like Tunisia, I must say that this process for the last five days since I have been in Egypt was enormously impressive and a tribute to the people of Egypt."
US Secretary of State Hillary Clinton congratulated Egypt on its "historic" election and said Washington looked "forward to working with Egypt's democratically elected government," according to a statement released by her spokeswoman.
On Thursday, the second and final day of voting, some of the main candidates accused each other of breaching electoral rules.
Mr Moussa and Mr Shafiq claimed false rumours were being spread that each was about to withdraw from the hotly contested race.
In a BBC interview, Mr Moussa launched an angry attack on his rival Mr Shafiq - who was appointed prime minister in the dying days of Mr Mubarak's rule - saying he represented the ideas of the old regime.
Mr Moussa denied what he described as "sinister rumours" that he was about to leave the race.
But in later interviews Mr Shafiq hit back, accusing Mr Moussa's campaigns of spreading similar rumours - and pointing out Mr Moussa's connections to the old regime. Mr Moussa served as a minister under Mubarak from 1991 to 2001.
Mr Shafiq, Mr Fotouh and Mr Mursi have all been accused of breaking rules requiring candidates keep silent on polling days - claims which the electoral commission said it would investigate.
Turnout appeared to vary across the country on Thursday, which authorities had declared a public holiday.
In Cairo, there were long queues at some polling stations but elsewhere, such as Alexandria and Suez, there were reports of slow voting.
Egypt's election commission estimated that about 50% of eligible voters had participated, reported official news agency Mena.
The military body that assumed presidential power in February 2011 - the Supreme Council of the Armed Forces (Scaf) - has promised a fair vote and civilian rule.
Until a new constitution is approved it is unclear what powers the president will have, prompting fears of friction with a military which seems determined to retain its powerful position.