Ian was in the Caribbean Sea on Sunday night, with maximum sustained winds of 60 mph, the National Hurricane Center said, but the storm was intensifying.
It was forecast to skirt western Florida on Wednesday and Thursday, according to the National Hurricane Center.
Ron DeSantis declared a state of emergency for all of Florida because of the threat, and the Florida National Guard activated 2,500 service members.
Tropical storm warnings covered the Cuban provinces of La Habana, Mayabeque and Matanzas, and Little Cayman and Cayman Brac were under a tropical storm watch.
The entire western coast of Florida is vulnerable to storm surge, said Rhome, of the National Hurricane Center.