Hurricane Gustav moved toward western Cuba on Saturday with winds of 230km per hour on its way to the Gulf of Mexico.
It is set to strike the Cuban mainland later in the day as a major hurricane.
Forecasters predicted Gustav would cross the Gulf of Mexico and hit central Louisiana on Tuesday with the same force that Hurricane Katrina delivered three years ago.
Officials in New Orleans say they could order a mandatory evacuation if Gustav holds to its current course.
The US National Hurricane Center said Gustav is now a Category 4 storm with winds of 230kph.
It may strengthen further on Sunday when it goes into the Gulf of Mexico, where offshore platforms produce 25% of US oil and 15% of its natural gas.
Thousands of people in Cuba have moved to shelters. Recent harvests of tobacco crops have been moved to safe places. Residents of Havana are being warned to seek protection.
The storm killed up to 77 people as it crossed the Dominican Republic, Haiti and Jamaica. No deaths had yet been reported from the Caymans, but power was out in much of the islands.
A tropical storm warning has been posted for the western Florida Keys.
Katrina hit the coast near New Orleans as a Category 3 hurricane on 29 August, 2005. Its storm surge broke through levees and flooded 80% of the city.
About 1,500 people were killed along the US Gulf Coast. Damage amounted to $US80 billion.
Hanna is coming
Meanwhile, another storm, Tropical Storm Hanna, is moving west in the Atlantic Ocean.
The hurricane center said it could be near hurricane strength by Sunday. A tropical storm watch has been issued for the southeastern Bahamas and the Turks and Caicos islands.