The report said authorities have moved 5 158 people from low-lying camps in the Rakhine state capital, Sittwe, to safer shelter.
It said displaced people were also moved in 10 other township in western Myanmar where communal violence flared last year between Muslims and Buddhists, taking hundreds of lives and leaving more than 100 000 people homeless.
It did not give the number of people evacuated in those locations.
The United Nations estimates that nearly 70 000 of those displaced are in low-lying areas along the coast that are highly susceptible to tidal surges and flooding.
In 2008, Cyclone Nargis killed more than 130 000 people in Myanmar.
Cyclone Mahasen is expected to hit neighboring Bangladesh on Thursday or Friday.
Spillover rain could cause landslides and flooding in Myanmar.
Aid groups have issued warnings for weeks that annual monsoon rains could wreak havoc for displaced people in their camps and spark disease outbreaks.
"The government has been repeatedly warned to make appropriate arrangements for those displaced in Rakhine state.
"Now thousands of lives are at stake unless targeted action is taken immediately to assist those most at risk," Isabelle Arradon, Amnesty International's deputy Asia Pacific director, said in a statement Monday.
The group acknowledged reports of some evacuations but said "several of the identified evacuation sites are within already established camps for internally displaced persons or fail to have adequate storm-ready structures, and storm warnings have not been provided to all at-risk displaced communities" outside of Sittwe.
"We need to be prepared for the worst," the United Nations' resident and humanitarian coordinator for Myanmar, Ashok Nigam, said Saturday.