Meanwhile in the city of Miami, anxious residents flocked to buy fuel and bottled water while television stations broadcast warnings about possible business shutdowns and power cuts. Schools in some areas were evacuated.
'Calm but alert'
As of 8pm EDT (00:00 GMT), Fay was about 169 km south of Naples and moving north at about 14 kph.
"Florida is very, very practiced at these things," Charlie Crist, the state's governor, said.
"The watch word for Floridians is be calm, but be alert."
The authorities also warned that flooding of up to 23cm was possible across mainland Florida.
"This is a broad, really diffuse storm. All the Florida Keys and all the Florida peninsula are going to feel the effects of this storm, no matter where the centre makes landfall,'' Stacy Stewart, a senior hurricane specialist at the National Hurricane Centre told AP.
"We don't want people to downplay this.''
Haiti bus deaths
The storm had already caused death and destruction across large parts of the northern Caribbean after heavy rains and winds battered Haiti and the Dominican Republic.
In Haiti, a bus plunged into a river swollen by rain from Fay, killing about 30 people, officials said.
In the Dominican Republic, more than 2,000 people were evacuated from their homes to shelters as the storm felled trees, damaged hundreds of homes and power lines, cutting supplies to more than 15,000 homes, according to reports.
Five others were killed in Haiti and the Dominican Republic in flood-related accidents.
Cuban officials had also ordered the evacuation of low-lying parts of Havana, but the storm produced only a mild breeze and intermittent showers there.