Election authorities said Monday final results aren't due until Tuesday, but Mosaique, the Tunisian radio station, announced Ennahdha, Renaissance in English, was pulling away in the vote count, The New York Times reported.
Party officials speculate taking 42 percent to 55 percent of the seats in the assembly.
Euronews said there were long lines of people outside almost all polling stations across Tunisia waiting to cast their votes to elect an interim assembly to run the country and draft a new constitution. About 4.1 million people had registered to vote out of more than 7 million who were eligible in the nation of about 10.4 million people.
The people of Tunisia toppled President Zine el-Abidine Ben Ali, becoming among the first in the region to cast off longtime rulers in what has become known as the Arab Spring.
Euronews said the since the Ennahda party is only expected to win a plurality of the 217 seats, it will need to form a coalition.
Tunisia Live reported the top elections official, Kamel Jendoubi, said while the polling went smoothly without major problems or interruption, there were some electoral infractions.
"We have information about meetings of large number of people in front of voting places shouting slogans," he said, adding "short messages were sent by mobile phones" and there had been some "soft intimidation efforts to influence voters."
None of the violations were serious enough to impact the final results, however.
"People's faces sometimes tell more than the whole procedure. There's serenity and joy at the same time," Gabriele Albertini, head of the European Parliament delegation, said. "A few times I saw people -- young and not so young -- with the tears in their eyes."
U.S. President Barack Obama congratulated Tunisians, calling the elections "an important step forward."
"The United States reaffirms its commitment to the Tunisian people as they move toward a democratic future that offers dignity, justice, freedom of expression, and greater economic opportunity for all," he said in a statement.
Marwen Hamadan, a 23-year-old architecture student, said he had voted for Communist candidates.
"I don't want to live with Islamic ideology," he said. "Tunisia is a diverse civilization, it is a mix of political and religious opinions. [Ennahda] will be a dictatorship in another form -- before it was a political one and I worry that Ennadha would impose a religious one."
Tunisia was one of the first Muslim countries to rebel against religious and autocratic rule in the political wave that began in January. The unrest led to further demonstrations in Syria, Yemen, Bahrain and most recently in neighboring Libya, where dissent evolved into civil war and saw ousted dictator Moammar Gadhafi killed.