Authorities in Florida have urged residents to "make their preparations" as Tropical Storm Ian strengthened into a Category 1 hurricane early on Monday.
Ian was forecast to move northwest through the Caribbean, near Cuba and the Cayman Islands, and reach the state's west coast by Wednesday.
The hurricane's exact path is uncertain but Governor Ron DeSantis warned of "broad impacts throughout the state".
Residents have been seen stocking up on food, water, medicine and fuel.
The National Hurricane Center (NHC) said it expects "rapid strengthening" from Ian over the next 48 hours, with maximum sustained winds of 75mph (121kmph).
An NHC official said Cuba could experience "extreme hurricane force winds, life threatening storm surge and heavy rainfall" later on Monday.
Classes have been cancelled in parts of the island, and evacuations have been planned in a handful of western provinces, including near the country's tobacco region.
Authorities in the Cayman Islands, a British territory, are working to provide plywood, sandbags and other supplies to local residents,
"We must prepare for the worst and absolutely pray and hope for the best," Premier Wayne Panton said in a video on Sunday.
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 on Sunday.
Local officials in her municipality and others like 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 approaching hurricane makes landfall.
Over the weekend, the governor declared a state of emergency for all of Florida and activated 2,500 National Guard troops to assist with relief efforts.
In his executive order on Sunday, Mr DeSantis said Ian posed a risk of "dangerous storm surge, heavy rainfall, flash flooding, strong winds, hazardous seas, and isolated tornadic activity".
The White House has also made its own emergency declaration, which will help federal and state officials coordinate disaster relief and assistance.
Forecasters have said Cuba could see up to 10 inches (25cm) of rain, while Jamaica and the Cayman islands could receive between 3-6 inches(8-15cm).
The Caribbean is still reeling from the effects of Hurricane Fiona, which tore through the region last week.
Moving northward to parts of the Atlantic Canada coastline and eastern Quebec, Fiona claimed two lives, washed homes into the sea and downed power lines over the weekend.