It said authorities acted after receiving information the suspects were preparing a spate of attacks against state and tourist institutions and churches.
A roadside bomb hidden in a wall exploded as the bus carrying 14 Vietnamese tourists went by.
Commenting on the terrorist attack that killed two Vietnamese tourists and injured ten others, Madbouly said that the tourist bus drove away of the police forces that were supposed to secure its path.
The interior ministry confirmed the death of two of the tourists, and the state prosecutor's office later said a third had died.
On Friday, a roadside bomb hit a tourist bus in an area near the Giza Pyramids, killing two Vietnamese tourists and wounding 12 others.
Madbouli was set to visit the injured tourists in hospital after following up with his ministers on the incident, a government statement said, as reported by AFP.
What do we know of Friday's attack?
Egypt's interior ministry has said at least 40 fighters have been killed on Saturday in response to an attack on a tourist bus in which four people killed the night before.
Egypt's vital tourism industry has been struggling to recover from terror attacks and domestic instability that have hit the country in recent years.
The Pyramids of Giza are the only surviving structure of the seven wonders of the ancient world and a major tourist draw attracting visitors from across the globe.
In 2015, ISIS claimed responsibility for a blast that downed a Russian airliner and killed all 224 people aboard shortly after it took off from the Egyptian holiday resort of Sharm El Sheikh.
The blast and the subsequent police raids come as Egypt battles a persistent jihadist insurgency in the North Sinai, which surged after the 2013 overthrow by the army of Mubarak's Islamist successor Mohamed Morsi.