The US has expressed renewed concern about violence in Egypt sparked by President Mohammed Mursi's moves to consolidate power.
The Brotherhood hopes to end the crisis by replacing Mursi's controversial decree giving himself dictatorial powers with an entirely new constitution that would need to be approved in a popular referendum, a Brotherhood official said.
It is a gamble based on the Islamists' belief that they can win the referendum: they have won all elections held since Hosni Mubarak was toppled from power.
But the move seemed likely to deepen divisions that are being exposed in the street.
But as Mr Mursi's opponents staged a sixth day of protests in Tahrir Square, critics said the Islamist-dominated assembly's bid to finish the constitution quickly could worsen matters.
Two people have been killed and hundreds injured in a countrywide protest set off by Mursi's decree.
The constitution is one of the main reasons Mursi is at loggerheads with non-Islamist opponents. They are boycotting the 100-member constitutional assembly, saying Islamists are imposing their vision on Egypt.
The assembly's legal legitimacy has been called into question by a series of court cases demanding its dissolution.
Its popular legitimacy has been hit by the withdrawal of members including church representatives and liberals.
"If you are upset by the decree, nothing will stop it except a new constitution issued immediately," Hossam el-Gheriyani, the assembly speaker, said at the start of its latest session in Cairo.