The storm threatens about 60,000 people in camps in low-lying areas of the country near the Andaman Sea, The New York Times reported Saturday.
Vivian Tan, a spokeswoman for the U.N. High Commissioner for Refugees, said the agency was "very concerned."
"We are working around the clock, trying to get as many people out of low-lying areas and into decent shelters," she said.
The cyclone could make landfall at the Bangladesh port city of Chittagong Wednesday, the U.S. Navy Marine Meteorology Division forecast on Saturday. Winds and heavy rains would hit Rakhine state, the area of Myanmar where the camps are located in muddy fields.
More than 150,000 died in 2008 when Myanmar's Irrawaddy river delta was overcome by heavy rains.