Two weeks after the army ousted President Mohammed Morsi, Mansour said in his first public address that his government will protect the country against those who seek chaos and violence.
"We are going through a critical stage and some want us to move towards chaos and we want to move towards stability. Some want a bloody path," the interim leader said, speaking in a pre-recorded message aired on state TV.
"We will fight a battle for security until the end. We will preserve the revolution."
Meanwhile, Morsi's Muslim Brotherhood called for mass rallies on Friday to demand his reinstatement.
The group refused to join the new interim cabinet unveiled this week.
Morsi was ousted on 3 July in what his supporters have said was a military coup. The army, however, says it was fulfilling the demands of the people after mass anti-Morsi protests.
Muslim Brotherhood supporters are demanding Morsi's reinstatement and have been holding a round-the-clock vigil outside the Rabaa al-Adawiya mosque, in the east of the capital, and at Cairo University in Giza.
Brotherhood officials said on Thursday they would accept EU mediation in talks to resolve the crisis but that they would continue their protests.
"Restoring legitimacy is non-negotiable," Brotherhood spokesman Gehad el-Haddad said after meeting with EU envoy Bernardino Leon.
It is the first time the group has proposed such negotiations since the overthrow of Morsi earlier this month, although both the Brotherhood and the EU cautioned that the proposal was at a very early stage.
The Muslim Brotherhood rejected Prime Minister Hazem al-Beblawi's transitional cabinet, which was sworn in earlier this week.
"It's an illegitimate government, an illegitimate prime minister, an illegitimate cabinet. We don't recognize anyone in it," Gehad el-Haddad said.
Dozens of people have died and hundreds more have been injured in deadly clashes between Morsi supporters and security forces over the last two weeks.
SHI/SHI