A member of Burundi's independent electoral commission, Clotilde Niragira, told VOA that 91 percent of the more than 2.5 million votes counted so far support the constitution.
Ms. Niragira said her commission has four days to announce the final count.
She said that, despite difficulties at some of the polling stations, the results reflect the will of the people.
"A lot of people had already voted 'yes'. If we have few problems in that period, they cannot influence the result of the referendum," he said. "I think that people will accept the results very well."
Burundi's ambassador to Kenya, Stanislas Nsabuwanka, praised what he called the calm and democratic process and said the new constitution represents a major turning point in his country.
"We know the rule - how people must cohabitate, how they must share everything," he said. "The constitution show[s] us how we can cohabitate between ethnic group(s), how we could organize our institution[s], and show us how we could build our nation."
Monday's referendum follows the requirements of a peace deal that was signed in Tanzania almost five years ago to end more than a decade of fighting between the Hutu and Tutsi ethnic groups.
The conflict has claimed about 300,000 lives.
The peace deal created a three-year transitional government that was supposed to hand over power to an elected government last year after the acceptance of a constitution that would, among other things, share power between Hutus and Tutsis. But the referendum had been delayed several times.
Mr. Nsabuwanka said that, now that the population has accepted the constitution, elections will be held sometime next month.
The head of the Tutsi party Rally for Democracy and Economic and Social Development, Joseph Nzeimana, charged that the referendum results are not valid because of a number of irregularities such as people voting twice, people voting both "yes" and "no", and voters being pressured by politicians.
But Mr. Nzeimana said he and his party would not contest the results.
"The best thing is to go [ahead] politically in order to get stability and stop the war," he said. "That is the first thing the politician has to do. We have to be a bit quiet."
Tutsis make up about 15 percent of Burundi's population, yet they dominate the army and political sphere, an imbalance that was a major cause of the war.
The constitution calls for a 50-50 split in the senate and a 60-40 split between Hutus and Tutsis in the National Assembly.
Some Tutsi political parties are unhappy with the constitution, saying that it does not adequately guarantee that Tutsis would continue to exercise political power in the country.