Its refugee agency has said aid camps were at “breaking point” and were struggling to cope with the influx of people.
UNHCR spokeswoman Vivan Tan said “The numbers are very worrying. They are going up very quickly,”
Hundreds of Rohingya Muslims are thought to have been killed in violence involving the Myanmar military.
The unrest broke out on 25 August when Rohingya insurgents attacked Myanmar police posts.
In response Myanmar security forces started days of “clearing operations” they claimed were needed tackle terrorism.
Burmese officials and Rohingya insurgents have both accused one another of committing atrocities in the past week.
Aid workers in Bangladesh say established camps that have been housing Rohingya since the 1990s are already operating beyond capacity.
The UNHCR says more land is needed to set up new camps to look after the refugees, many of whom are arriving hungry, traumatised and in need of medical assistance.
“Most have walked for days from their villages hiding in jungles, crossing mountains and rivers with what they could salvage from their homes,” the agency said.
“An unknown number could still be stranded at the border.”
Many of those arriving at the camps have told stories of their homes being set on fire and soldiers firing indiscriminately around their villages in Rakhine state.
The Rohingya have faced years of discrimination in Myanmar, also known as Burma, which has a majority Buddhist population.
There has been international condemnation of the treatment of the Rohingya in Myanmar.
Downing Street has called on Myanmar’s leader, Aung San Suu Kyi, to take action to stop the violence.
Ms Suu Kyi is facing a growing chorus of criticism for not speaking out for the minority that has long complained of persecution.