Michel Kafando told the Security Council that consultations are expected to be held in the coming days to relaunch talks between the government and the opposition, under the auspices of East African leaders.
"We expect a strong signal from the authorities in support of relaunching the inter-Burundian dialogue under the aegis of the East African Community," said Kafando, a former president of Burkina Faso who became UN envoy last year.
The United States and France also called for dialogue, but Russia warned against meddling in Burundi's domestic affairs, while China stressed that the situation was "generally stable."
"The priority should be to increase humanitarian assistance to Burundi and support for its economic and social development," said China's representative at the meeting, Zhang Dianbin.
In power since 2005, Nkurunziza plunged his east African nation into crisis in 2015 when he circumvented a constitutional two-term limit, stressing his first term came after an election by parliament.
At least 1,200 people have died and 400,000 displaced in a crackdown on protests against his rule.
In a referendum held on May 17, a majority of voters backed constitutional changes that would extend the presidential term limit to seven years.
The opposition accused the government of waging a campaign of fear and intimidation ahead of the vote.
Nkurunziza has refused to hold talks with the opposition, which he accuses of staging an alleged coup.
The government has branded as a terrorist organization the umbrella opposition group the National Council for the Restoration of Arusha Agreement and Rule of Law, or CNARED.
Nkurunziza is the latest African leader to seek changes to the constitution in order to stay in power, along with Rwanda's Paul Kagame, Uganda's Yoweri Museveni, Cameroon's Paul Biya among others.
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