Mr Nkurunziza has already served two terms as president
Sylvere Nimpagaritse, vice-president of Burundi?s constitutional court, told the AFP news agency that the court?s judges have faced ?death threats?.
President Pierre Nkurunziza?s third-term bid has led to more than a week of deadly protests.
Under the constitution, presidents can only be elected to two terms in office.
Mr Nkurunziza?s allies say his first term does not count as he was appointed by parliament.
The decision over whether the president?s third-term bid is legal is set to be made by the constitutional court.
Mr Nimpagaritse says that most of the court?s judges think it is unconstitutional, but they are now under pressure to change their minds, reports AFP.
Mr Nkurunziza has been in power since 2005, when a 12-year civil war officially ended.