No official results in the parliamentary vote were out and one rival party said they doubted the Brotherhood's claim, but others provided estimates that appeared to back it.
The results, if confirmed and repeated when the rest of the country votes in successive rounds of the staggered six-week election process, would give Egypt's oldest Islamist group a powerful bloc in the assembly.
It could end up positioned for a power struggle with the generals who took over when Hosni Mubarak was ousted in a popular revolt in February.
Progress towards democracy in the most populous Arab nation will help shape a region convulsed by popular uprisings against autocrats who, like Mubarak, often enjoyed Western support in part for their role in fighting Islamist militancy.
Egypt's complex electoral system and staggered voting means it may be difficult to predict the precise breakdown of parliament until the end of staggered voting on January 11.
Two-thirds of 498 lower house seats will be allocated proportionally by party lists and one third to individuals. The Brotherhood's Freedom and Justice Party (FJP) said early indications showed it was ahead in both races.
In the party list race it was followed by the ultra-conservative Islamist al-Nour Party and the liberal Egyptian Bloc, it said in a statement.
An FJP source, who declined to be named, said the FJP-led list had won about 40 percent of the party-list votes so far.
The army council that has ruled since the uprising toppled Mubarak on February 11 has said it will retain powers to choose or dismiss a cabinet. But the FJP's leader said on Tuesday the majority in parliament should form the government.
The cabinet resigned last week amid demonstrations against army rule in which 42 people were killed. On Friday the generals picked Kamal al-Ganzouri, a Mubarak-era premier, to form a new cabinet, which he has said will be ready this week.
Monday and Tuesday's voting, the first of three rounds which will each be followed by run-offs, passed off mostly peacefully, but violence broke out on Tuesday night in Cairo's protest hub of Tahrir Square, where nearly 80 people were wounded.
A member of the liberal Egyptian Bloc alliance said the Brotherhood's FJP had won 40 to 50 percent of votes in Cairo, with his bloc running second with 20 to 30 percent.
But a much smaller liberal party, the Justice Party, said the Brotherhood's estimates were "very skewed." Spokeswoman Nora Soliman said the Brotherhood was strong in areas but added: "All the numbers they came out with are presumptuous and are designed to create momentum for the second round."
First-round results are expected to trickle out on Wednesday after a good turnout and only minor abuses. Western governments praised the high participation and the vote's smooth conduct.
The once-banned Brotherhood did not start the anti-Mubarak revolt, but the political shift since then has propelled it closer to power, even though the army has yet to step aside.
In an implicit challenge to the military's authority, the head of the FJP said parliament should form the government.
"A government that is not based on a parliamentary majority cannot conduct its work in practice," FJP head Mohamed Mursi told reporters in Cairo's working-class district of Shubra, adding that a coalition government would be best.
The generals, already under pressure from Egyptians angered by their perceived desire to hold onto military perks and power, may face a new challenge from a parliament flush with the popular legitimacy gained from a big turnout at the polls.
One member of the military council has said turnout would exceed 70 percent. The FJP's Mursi put it at 40 percent.
General Ismail Atman, an army council member, was quoted as saying the poll showed the irrelevance of the Tahrir protests.
The latest violence there erupted on Tuesday when unidentified youths tried to enter the square, a protest organizer said. Petrol bombs were thrown at protesters and guns were fired. Of the 79 wounded, 27 were taken to hospital, the state news agency said.
Criticizing the authorities, reformist politician Mohamed ElBaradei wrote on Twitter: "Thugs are now attacking the protesters in Tahrir. A regime that cannot protect its citizens is a regime that has failed in performing its basic function."