More than 5,800 Burundian refugees have crossed into Rwanda so far this month amid fears of violence before elections later this year, the UN refugee agency has said.
Failure to organise free and fair elections could prompt a return to deadly violence in Burundi, according to a top UN rights official.
Presidential elections are scheduled for June 26 and incumbent President Pierre Nkurunziza has not publicly said if he will seek a third term.
A dispute over whether Nkurunziza can run for a third term has led to a political crisis, said to be the worst since the 12-year civil war in the East African nation ended in 2005.
Opposition parties and dissenting members within the ruling CNDD-FDD party say Nkurunziza running for a third term would violate a 2000 deal that ended the conflict between Hutu and Tutsi ethnic groups.
Burundian opposition parties are discussing uniting behind a single candidate to improve their chances of defeating Nkurunziza should he run for re-election in June, an opposition official told Reuters news agency on Thursday.
"Discussions have started ... to see how we can pick one candidate in order to defeat Nkurunziza's ruling party," said Chauvineau Mugwengezo, spokesperson for the Alliance for Democratic Change (ADC-IKIBIRI), a grouping of five opposition parties.
Meanwhile, hundreds of Burundians have been crossing the border in neighbouring Rwanda daily, most of them children.
At least 800 of the refugees arrived in Rwanda on Tuesday, the UN refugee agency said in a statement e-mailed to the Associated Press news agency on Wednesday.
"We have ... observed a very sharp increase in arrivals in recent days, of which over 60 percent are children," said Saber Azam, the UN refugee agency's representative in Rwanda.
There is growing alarm among diplomats and observers "about the direction the country appears to be taking," Zeid Ra'ad al-Hussein, the UN human rights commissioner, said in Burundi's capital, Bujumbura.
"The country is at a crossroads," he said, warning that failure to hold free elections could return Burundi to what he called a "deeply troubled, tragic and horrendously violent past."
Also on Wednesday, the spokesman for UN secretary-general said Ban Ki-moon met with Burundi's interior minister, Edouard Nduwimana, on Tuesday and expressed concern about the rising political tensions. Ban urged that everyone be allowed to participate freely in the electoral process.
The Burundian refugees cite disappearances of relatives and pressure to pledge allegiance to the ruling party among reasons for fleeing their small East African country, said the UNHCR statement.
Other refugees cite fear of a youth militia as elections approach in Burundi, the UNHCR said.
That militia, known as Imbonerakure, is made up of members of the ruling party's youth league and is often accused of violating human rights in northern Burundi.
Human Rights Watch, the New York-based rights group, has accused the Imbonerakure of participating in the recent killing of at least 47 people in the northern province of Cibitoke.
The Cibitoke killings - between December 30, 2014 and January 3, 2015 - are "part of a broader pattern of extrajudicial executions" by Burundian security forces and the Imbonerakure, said the rights group.