The BBC’s Maud Jullien reports from the scene that police fired shots, and she saw two of the bodies being taken away.
Police denied killing the protesters, and said 15 of their officers had been wounded in a grenade attack.
The US’s John Kerry urged Mr Nkurunziza to abandon his re-election bid.
“We are deeply concerned about President Pierre Nkurunziza’s decision, which flies directly in the face of the constitution of this country,” the US secretary of state told reporters during a visit to Kenya.
The Red Cross says 12 people have been killed since protests broke out on 26 April after Mr Nkurunziza was nominated by the ruling CNDD-FDD party as its candidate for the June election.
Under Burundi’s constitution, presidents can only be elected to two terms in office, but Mr Nkurunziza’s allies say his first term does not count as he was appointed by parliament.