UN special envoy Said Djinnit told the council's 15 member countries that conditions were "not ripe" for elections, according to the Lithuanian ambassador, who holds the body's rotating presidency.
Discussions are underway to move the vote to a later date in June, according to Djinnit.
The council members "called on all parties to refrain from violence," Lithuanian envoy Raimonda Murmokaite said after the closed-door talks.
She said the council also "stressed the need to hold a credible, transparent, inclusive and peaceful electoral process and uphold the fundamental freedoms of opinion and assembly."
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Russia has blocked previous council initiatives seeking to take a position on Nkurunziza, saying the body should not get involved in the internal affairs of a sovereign state.
But Murmokaite insisted that the council meeting focused on the growing violence in Burundi. Protesters, meanwhile, called a one-day truce, saying the demonstrations would resume on Sunday.
US envoy Samantha Power warned of potential targeted sanctions from Washington on perpetrators of the violence.
She stressed that Washington considers Nkurunziza's efforts to seek a third term to be illegitimate, and urged him and his government to condemn the violence and call for restraint.
She highlighted the distribution of weapons to youths backing the ruling party, in what she called an "extremely disturbing" development.
Diplomats who attended the meeting, called by France, said Djinnit described the situation as tense in Burundi.