The Brotherhood, which has emerged as a powerful force after years of repression under ousted President Hosni Mubarak, has said it does not want a parliamentary majority, although rivals see it as well placed for a dominant position.
With secular politicians struggling to mount a challenge, Western investors are concerned about what a shift to an Islamic-leaning government would mean for Egypt, which relies on receipts from Western and other tourists and where tension between Muslims and the Christian minority have flared.
Liberal Egyptians in particular worry that the group could use for its own ends the second article of Egypt's constitution, which makes sharia, Islamic law, a main source of legislation.
Egypt's military rulers suspended the old constitution and introduced an interim one, but that article was unchanged.
Mr. Mursi, speaking in the group's new five-storey headquarters in Mokattem on the outskirts of Cairo, dismissed such worries.
"We want to engage in a dialogue not a monologue," he said. "The Brotherhood does not seek to control the parliament ... We want a strong parliament ... with different political forces."
But he said Islamic law could have a place in a civil state in Egypt, where about 10 percent of the 80 million population are Christians. "Islamic sharia guarantees the rights of all people, Muslims and non-Muslims," he said.
Mr. Mursi said he would stick by the Brotherhood's pledge not to field a presidential candidate or support any Brotherhood member running, as one has already said he will do.
"The group said it will not field a candidate for the presidency or support one if decides to do so independently," he said.
The Brotherhood's new offices are emblazoned with its emblem of crossed swords, a scene unimaginable in the Mubarak era when its members were rounded up in regular sweeps and it worked from two cramped apartments in Cairo.
Mr. Mursi, head of the engineering department in Egypt's Zaqaziq University, led the Brotherhood's parliament bloc in the 2000-2005 parliamentary session. The Brotherhood used to field its candidates as independents to skirt a ban on its activities.
The Brotherhood, which has spread deep roots in Egypt's conservative Muslim society partly through a broad social program, held 20 percent of seats in the 2005-2010 parliament.
It boycotted last year's vote because of accusations of rigging, which rights groups said had been a feature of all votes under Mubarak.
Mr. Mursi said an economic platform had not yet been drawn up as the party, formed in April, was still organizing itself.
But some secular politicians and other Egyptians are concerned that women and Christians could be sidelined and that alcohol could be banned, which analysts say is a concern as many tourists to Egypt are non-Muslims wanting a beach holiday and who might be deterred if alcohol is not served.
One in eight Egyptian jobs depend on tourism.
On Christians, he said: "We want everyone to be reassured ... that we want to see our Christian brothers elected in parliament ... We don't want one group to control the parliament, neither the Brotherhood nor anyone else."
Of the party's 9,000 registered members, he said 100 were Christian and 1,000 were women, adding that the party's deputy head, Rafik Habib, was a Christian.
When asked if the party could propose a law to prohibit alcohol, Mr. Mursi said such changes would be up to parliament to decide, not a single group, such as the Brotherhood.
"The Egyptian constitution is not the constitution of the Brotherhood but ... of the Egyptian people," said Mr. Mursi, adding that the constitution "says Egypt's legislation is based on the principles of sharia, and not its details."