The move brings to an end weeks of sit-ins held in protest at the ousting of Mursi by the military on 3 July.
The two camps are at Rabaa al-Adawiya mosque in the east of Cairo and in Nahda Square in the west.
The number of protesters killed so far in the ongoing clashes is said to range from 15 to as many as 30. Hundreds have been injured. According to the Interior Ministry, at least two members of the security forces have been killed.
A statement from Egypts interior minister says a safe exit from the camps is available for protesters. It says it does not wish to spill a single drop of blood and is committed to deal with the situation according to the legal rules and procedures accepted.
The violence follows attempts by the EU and the Gulf countries to broker an agreement between the military-backed government and the protesters. A reconciliation agreement was presented by the chief cleric at Al-Azhar University, an institution highly respected by Sunni Muslims. This was rejected, however, by the Muslim Brotherhood and its political party the Freedom and Justice party (FJP).
The Muslim Brotherhood continue to demand the reinstatement of former president Mursi while the current interim government remains committed to pushing through amendments to the constitution and running fresh parliamentary and presidential elections in early 2014.
In a statement, the FJP show no signs of seeking a compromise with the interim government. We will defeat the bloody coup with our non-violent persistence, it said.
With no immediate prospect of a compromise agreement between the opposing parties, the impact of the political instability continues to damage Egypts economy.
Private capital inflows are likely to dwindle and central bank reserves will remain dependent on foreign donors. Growth hence will struggle to rise from current anaemic levels, says Raza Agha, chief economist, Middle East and Africa, at VTB Capital based in the UK, in a statement.