The Bangladeshi authorities have raised the danger level to seven out of 10 for low-lying areas around Chittagong and the coastal district of Cox's Bazar.
It is estimated that the cyclone, heading north-east through the Bay of Bengal, will reach land on Thursday.
Burma is also threatened and evacuation efforts are under way there.
Tens of thousands of Rohingya Muslims living in camps in low-lying areas of Burma's Rakhine state are feared to be at risk.
They were displaced by ethnic violence last year and many are reluctant to move from the camps.
BBC Weather update on the progress of Tropical Cyclone Mahasen and the tracks it could take
Hla Maung said he lost his mother and two young daughters during the clashes between Muslims and Buddhists.
"I lost everything ... I don't want to go anywhere. I'll stay here. If I die, I want to die here," he said.
The country's National Planning Minister, Tin Naing Thein, said more than 150,000 people had been relocated to higher ground. The government said a fifth of those were Rohingyas.
'Life-threatening'
The Bangladesh Meteorological Department said the storm could create a surge 2m (6.5ft) high in coastal districts and residents are being urged to make for cyclone shelters.
The airports in Cox's Bazar and Chittagong have been shut until the danger subsides.
The United Nations Office for the Co-ordination of Humanitarian Affairs said Cyclone Mahasen appeared to have weakened to a Category One storm.
But the UN still described it as "life-threatening" for 8.2 million people in Bangladesh, Burma and north-east India.
Though the storm has not made landfall in Sri Lanka, the associated heavy rain caused floods and mudslides which killed at least seven people, according to the country's Disaster Management Centre.
At least 50 Rohingya Muslims were feared drowned on Tuesday when boats evacuating them from the path of the cyclone capsized off western Burma.
More than 100,000 people died in 2008 when Cyclone Nargis devastated many of Burma's coastal villages.