Myanmar’s opposition National League for Democracy says it is confident of victory in the first openly contested national election in 25 years.
An NLD spokesman said it expected to win about 70% of seats. Party leader Aung San Suu Kyi said: “I think you all have the idea of the results.”
Official results have been released for just 12 seats, all won by the NLD.
The military-backed Union Solidarity Development Party (USDP) has been in power since 2011.
“We are on track to win more than 70% of seats around the country, but the election commission has not officially confirmed yet,” NLD spokesman Win Htein told AFP news agency. The 12 seats announced so far are all in Yangon.
The acting chairman of the USDP has told BBC Burmese that he has lost his own seat in the constituency of Hinthada to the NLD – seen as a key indicator of election results.
“We have to find out the reason why we lost,” U Htay Oo said. “However, we do accept the results without any reservations. We still don’t know the final results for sure.”
Earlier, Ms Suu Kyi addressed a crowd at the NLD’s headquarters in Yangon, urging them to be patient.
A quarter of the parliamentary seats are reserved for the army, and for the NLD to have the winning majority it will need at least two-thirds of the contested seats.
The BBC’s Jonah Fisher in Yangon says that while this is a very big ask, it is by no means impossible if the party, which is popular in urban areas, manages to win seats in rural areas which tend to be dominated by ethnic minorities.
But Ms Suu Kyi cannot become president because the constitution bars anyone with foreign children from holding the post. Her two sons, with her late husband, are British.
Our correspondent says that if the NLD win, it will face difficulties in changing the constitution on its own as the document still gives the military considerable power, and the party would most likely nominate someone else to be president. Ms Suu Kyi has said she would be “above the president”.
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Tens of thousands of officials and volunteers have been counting the votes, first in each of the 50,000 polling stations, then tallying them in constituency offices of the Election Commission.
In one of the earliest and most significant known results, the ruling party’s Shwe Mann, who is Speaker of the lower house of parliament, conceded defeat in his constituency to the NLD candidate.
The full results will not be known for at least a few days, and the president will only be chosen in February or possibly later.
International observers say the voting process was generally smooth, with some isolated irregularities.
And hundreds of thousands of people – including minority Rohingya Muslims – were also denied voting rights, raising concerns about the fairness of the poll.
US Secretary of State John Kerry hailed the elections as a step towards democracy, but added that they were far from perfect.
Analysis: Jonathan Head, BBC News, Yangon
After the euphoria of a historic election day, now the potentially fractious count.
Already the opposition NLD is complaining about the way large numbers of votes cast in advance of polling day were brought into the count late, apparently favouring the incumbent USDP.
The Election Commission has also delayed its first official announcement by several hours, although this is probably down to predictable logistical challenges.
Human rights groups have already criticised the Election Commission’s lack of independence. Election monitors have raised concerns over the absence of agreed procedures for resolving disputed results.
So a number of big questions still hang over this momentous turning point for Myanmar.
Will the official results give Aung San Suu Kyi’s party the two-thirds of contested seats it needs to form the next government? If they don’t, will she accept the result? Her supporters are already convinced they must have won.
Will the inevitable flaws in the electoral process be deemed small enough not to have compromised the result?
And how will Ms Suu Kyi and the military, historic foes, manage the protracted negotiations which must precede the formation of the next government whatever the election result?
About 30 million people were eligible to vote in Sunday’s election in Myanmar, which is also known as Burma.
Turnout has been estimated at about 80%, in what were the first national elections since a nominally civilian government took power in 2011
Media captionEager voters formed queues at one polling station
“I’m really happy because from what I heard the NLD is winning. I couldn’t sleep until 11 or 12 because I was looking everywhere for results,” San Win, a 40-year-old newspaper vendor, told the AP news agency.
More than 6,000 candidates from more than 90 parties were vying for parliamentary seats.