The Interior Ministry did not immediately confirm the death toll, but said in a statement that a number of detainees had tried to escape from a prison on the outskirts of Cairo and had taken a police officer hostage.
In subsequent clashes, the ministry said an undisclosed number of people had died from inhaling tear gas rounds. It added that the officer was freed but badly wounded.
However, offering a different explanation, a legal source told Reuters that the Brotherhood followers had suffocated in the back of a crammed police van while being taken to prison.
More than 800 people have died in Egypt since last Wednesday in clashes pitting backers of deposed Islamist President Mohamed Mursi and security forces. Police have said they have arrested more than 1,000 Brotherhood sympathizers in recent days as part of a broad crackdown aimed at ending weeks of political turmoil.