Hurricane Ian has been churning across Cuba as a category three storm with wind speeds of up to 205km/h.
Hundreds of thousands of residents in the west of the island were left without power because of storm damage.
Forecasters have warned of the risk of storm surges, flash floods and mudslides.
Ian is expected to strengthen en route to Florida. Florida Governor Ron DeSantis spoke of the threat of a “major disaster”.
As of 12:00 GMT on Tuesday, the centre of the storm was located about 15km (10 miles) north-east of the city of Pinar del Rio in Cuba, moving north at nearly 19km/h, the US National Hurricane Center (NHC) reports.
It said Cuba could see up to 30cm (12in) of rain from Hurricane Ian. Some 38,000 people were evacuated in the province of Pinar del Rio, according to AFP news agency.
Cuban authorities declared emergencies in six areas, with forecasters warning of storm surges on the coast, along with flash floods and mudslides.
Hurricane conditions are expected along the west coast of Florida on Wednesday.
The governor said there could be “broad impacts throughout the state”, while residents have been stocking up on food, water, medicine and fuel.
“This is a really big hurricane at this point,” he said at a Monday news conference, urging residents to “remain calm”, but “do what you need to be prepared”.
Some parts of the US state have not seen a hurricane of this magnitude in about a century.
The governor declared a state of emergency for all of Florida over the weekend and has activated 5,000 National Guard troops to assist with relief efforts.
Mr DeSantis said Ian posed a risk of “dangerous storm surge, heavy rainfall, flash flooding, strong winds, hazardous seas and isolated tornadic activity”.
Along Florida’s Tampa Bay coast, grocery store shelves were quickly cleared of basic necessities and there were long queues at gas stations.
“It’s never too early to prepare,” Tampa Mayor Jane Castor tweeted.
The Tampa area could receive its first direct hit from a hurricane since 1921, officials say, and may see 3m (10ft) of storm surge cause flooding along the coast.
Local officials in Tampa, Miami and Fort Lauderdale are distributing free sandbags to help residents protect their homes from flooding.
Meteorologists have said flash flooding is possible by Tuesday in the Florida peninsula and Florida Keys as the hurricane approaches.