General Godefroid Niyombare, a powerful former intelligence chief, announced the coup attempt hours after the president left for neighbouring Tanzania for talks with regional leaders on ending the crisis.
"President Pierre Nkurunziza is removed from office, the government is dissolved," General Niyombare said in the radio broadcast.
"All people are asked to respect the lives and property of others," he added.
Niyombare is a highly respected figure who was sacked from his position as the central African nation's powerful chief of intelligence in February.
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He said he would form a "committee for the restoration of national harmony," a temporary body whose "mission, among others, is the restoration of national unity... And the resumption of the electoral process in a peaceful and fair environment."
There was no immediate reaction from Nkurunziza, who was in neighbouring Tanzania to meet with leaders of the five-nation East African Community (EAC) -- made up of Kenya, Rwanda, Tanzania and Uganda as well as Burundi -- who are trying to mediate an end to the crisis.
Weeks of clashes between security forces and demonstrators have raised fears of a return to widespread violence in Burundi, which is still recovering from a brutal 13-year civil war that ended in 2006.
The African Union, European Union and United States have condemned the actions of Nkurunziza, a former rebel leader from the Hutu majority who has been in power for a decade.
Despite intense international pressure, Nkurunziza has repeatedly rejected international calls to end his bid for a third term.
But he argues that his first presidential term did not count as he was elected by parliament, not directly by the people.
Asked to rule on the issue, the constitutional court found in his favour but not before one of the judges fled the country claiming its members were subject to death threats.
More than 50,000 Burundians have also fled into neighbouring nations in recent weeks, with the UN preparing for thousands more to come.