Egypt is under a state of emergency as the military-backed interim leaders crack down on Islamists opposed to the army’s ousting of Mohammed Morsi.
Almost 900 people have been killed in the country since Wednesday.
Among them are 36 Islamist protesters who died in a prison van in Cairo on Sunday.
The UN’s human rights agency OHCHR has said the deaths of the detainees are “deeply disturbing and need to be fully investigated”.
Spokeswoman Liz Throssell said all those detained must be “treated humanely” and according to international law, and appealed to Egypt to allow UN human rights officers in to assess the situation.
European Union foreign ministers will meet on Wednesday to decide whether to cut some of the billions of euros in aid pledged to Egypt.
Hundreds of members of the Muslim Brotherhood – the movement from which Mr Morsi comes – have been detained, including Mr Badie’s deputy, Khairat al-Shatir, who was arrested in the days following Mr Morsi’s overthrow on 3 July.
Mr Badie, 70, had initially been a prominent figure at the Brotherhood’s protest camps in Cairo, but went into hiding as the military-backed interim government increased its efforts to shut down the protests.
He is facing charges of inciting violence and murder over the deaths of eight anti-Brotherhood protesters outside the movement’s headquarters in Cairo last June.
Officials said he had been detained in a flat in Nasr City in north-east Cairo, near the site of one of the protest camps bloodily broken up by security forces last week.
The interior ministry released footage of him apparently taken shortly after his arrest.
The Brotherhood’s political wing, the Freedom and Justice Party, said Mahmoud Ezzat – currently the movement’s deputy leader and who is also subject to an arrest warrant – would temporarily take over from Mr Badie as “supreme guide”.
Egypt is officially observing three days of mourning for 25 policemen killed by suspected Islamist militants near the Rafah border with Gaza in Sinai on Monday, in one of the deadliest attacks on security forces in several years.
In a separate incident, another police officer was killed in the north Sinai town of el-Arish.
Sinai is home to a range of militant groups, some linked to al-Qaeda, and while state media have not connected the killings to the Brotherhood, it has added to the sense of crisis, says the BBC Jeremy Bowen.
Attacks by Islamist militants on the Egyptian security forces have surged in the north of the peninsula since 2011 – and have become almost daily in recent weeks.
Meanwhile, Egyptian prosecutors have added a further 15 days to Mr Morsi’s detention while they investigate fresh allegations against him.
He has reportedly been accused of complicity in acts of violence against protesters outside the presidential palace last December.
His detention had already been extended by 30 days in a separate case on Thursday.
A lawyer for former President Hosni Mubarak has said he hopes his client could soon be released once cleared of a final corruption charge.
Mubarak is facing a retrial for corruption and complicity in the deaths of protesters during the 2011 uprising which removed him from power after 30 years.
Correspondents say his release, if it happens, would be seen by many as a sign the military is rolling back the changes that flowed from the uprising.