BBC
Official results are not expected until Thursday, but the Freedom and Justice Party said it had won 36 of the 56 seats awarded to individual candidates.
The Islamist group won almost 37% of the vote in earlier polling, which awarded seats according to party lists.
It was the first ballot since President Mubarak was ousted in February.
Voting in the remaining two-thirds of electoral districts is scheduled to take place later this month and in January 2012.
'Rainbow coalition'
Under Egypt's electoral system, two-thirds of the 498 elected seats in the lower house of parliament, the People's Assembly, will be picked through proportional representation, using lists drawn up by parties.
The remainder are decided by a first-past-the-post-system, with individual candidates required to win more than 50% of the votes.
Twenty-four seats were contested by the FJP and the ultra-conservative Salafist al-Nour Party, which came second with 24% of the vote.
Al-Nour party spokesman Yousseri Hamed told the Associated Press that it had won at least five additional seats in the run-offs.
On Tuesday, the Muslim Brotherhood's General Guide, Mohammed Badie, said the FJP would form a broad coalition if it won the elections.
"We will not rule Egypt alone. Parliament will include all the colours of the rainbow that must agree on one direction, one goal," he told the private television station, al-Mehwar.
Mr Badie also said he would not seek to get into a power struggle with the ruling council over the government, which assumed the president's powers after Mr Mubarak stood down.
"We must live in harmony, not only with the military council, but with all of Egypt's factions, or else the conclusion is zero," he added.
Many supporters of liberal and secular parties are concerned that Islamist parties, including the FJP and al-Nour, will have too much power in the next parliament, which will select a 100-member panel to draft a new constitution that will be put to a referendum before a presidential election in June.
Also on Tuesday, Prime Minister Kamal al-Ganzouri revealed the majority of the names in his new cabinet.
Many ministers were set to keep their portfolios, but Mumtaz al-Said was confirmed as the new finance minister, state television reported.
The new interior minister would be announced when the cabinet took the oath of office on Wednesday, it said.
Mr Ganzouri, 78, who served as prime minister under Mr Mubarak, was tasked with forming a government of "national salvation" after deadly clashes between protesters and security forces last month.
The prime minister also said the military council would issue a decree giving him more decision-making powers, "except those concerning the judiciary and armed forces".
The military has been accused of trying to safeguard its interests from civilian oversight and slowing down the transfer to civilian rule.
The Muslim Brotherhood's leader says it will form a broad coalition if it wins the elections
The Muslim Brotherhood has said its political party has won a majority of run-off contests in the first round of Egypt's parliamentary election.
The Islamist group won almost 37% of the vote in earlier polling, which awarded seats according to party lists.
It was the first ballot since President Mubarak was ousted in February.
Voting in the remaining two-thirds of electoral districts is scheduled to take place later this month and in January 2012.
'Rainbow coalition'
Under Egypt's electoral system, two-thirds of the 498 elected seats in the lower house of parliament, the People's Assembly, will be picked through proportional representation, using lists drawn up by parties.
The remainder are decided by a first-past-the-post-system, with individual candidates required to win more than 50% of the votes.
Continue reading the main story
“Start Quote
Mohammed Badie Muslim Brotherhood General GuideWe must live in harmony, not only with the military council, but with all of Egypt's factions”
Only four seats were won outright in last week's first round, leaving 52 to be decided in run-offs on Monday and Tuesday.
Al-Nour party spokesman Yousseri Hamed told the Associated Press that it had won at least five additional seats in the run-offs.
On Tuesday, the Muslim Brotherhood's General Guide, Mohammed Badie, said the FJP would form a broad coalition if it won the elections.
"We will not rule Egypt alone. Parliament will include all the colours of the rainbow that must agree on one direction, one goal," he told the private television station, al-Mehwar.
Mr Badie also said he would not seek to get into a power struggle with the ruling council over the government, which assumed the president's powers after Mr Mubarak stood down.
"We must live in harmony, not only with the military council, but with all of Egypt's factions, or else the conclusion is zero," he added.
Continue reading the main story
Egypt's complicated vote
- Three separate polls over coming months
- Elections to 508-member People's Assembly (lower house) - 28 Nov-10 Jan 2012
- Elections to 270-strong Shura Council (upper house) - 29 Jan-11 March 2012
- Presidential elections due mid-2012
- Two-thirds of members for both houses elected by PR
- One-third chosen by first-past-the-post system
- Provinces divided into three groups, voting on different dates
- More than 40 political parties compete, fielding more than 10,000 candidates
The secular Egyptian Bloc came third in the first round with 13.4% of the vote, followed by the liberal Wafd Party with 7.1% and the moderate Islamist Wasat Party with 4.3%. The Revolution Continues, a group formed by youth activists behind the uprising that ousted Mr Mubarak in February, won 3.5%.
Also on Tuesday, Prime Minister Kamal al-Ganzouri revealed the majority of the names in his new cabinet.
Many ministers were set to keep their portfolios, but Mumtaz al-Said was confirmed as the new finance minister, state television reported.
The new interior minister would be announced when the cabinet took the oath of office on Wednesday, it said.
Mr Ganzouri, 78, who served as prime minister under Mr Mubarak, was tasked with forming a government of "national salvation" after deadly clashes between protesters and security forces last month.
The prime minister also said the military council would issue a decree giving him more decision-making powers, "except those concerning the judiciary and armed forces".
The military has been accused of trying to safeguard its interests from civilian oversight and slowing down the transfer to civilian rule.
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