The devices were planted yesterday, less than 20 meters (yards) away from the walls of the Ittihadiya palace in the upscale Heliopolis district in eastern Cairo, in what appeared to be a serious security breach in the heavily policed area.
It was not immediately clear whether President Abdel-Fattah el-Sissi, who as army chief ousted Morsi last summer, was inside the palace when the explosions occurred.
He did not elaborate, but the government has drafted an anti-terrorism bill that has been delayed by criticism from human rights organisations and liberal politicians as well as the absence of an elected parliament.
"Black terrorism is still trying to stand between the Egyptians' will, their hopes and aspirations, a terrorism that knows no religion or nation," el-Sissi said in a pre-recorded speech. "I promise God, their families and their pure souls that the state will avenge their deaths justly and fast."
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"We will continue to defend our religion and we will deal firmly and forcefully with reactionary forces," he said.
The government holds Morsi's Muslim Brotherhood group responsible for the violence that followed his ouster, claiming is seeking to destabilise the new order in cooperation with radical Islamists.
In a stunning reversal of fortunes, the government declared the 86-year old group, which rose to power after Egypt's 2011 uprising, a terrorist organisation. It has arrested thousands of its members and froze the assets of its senior leaders.
The Brotherhood denies the charges, and says the accusations are part of a political crackdown seeking to defame the group.