Preliminary results are expected later on Monday.
But NLD leader Aung San Suu Kyi has hinted at victory saying: "I think you all have the idea of the results."
The military-backed Union Solidarity Development Party (USDP) has been in power since 2011.
Ms Suu Kyi was speaking to reporters and supporters at the NLD's headquarters in Yangon earlier on Monday. She also said it was "still a bit early to congratulate our candidates who will be the winners".
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."
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".
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.
Source: BBC