Four people were killed in the Dominican Republic due to the heavy rains, while thousands were evacuated, local media reported.
In Haiti, where a bus plunged into a river reportedly killing dozens, the heavy rains killed seven people and left another three missing, civil protection officials told AFP.
Maximum sustained winds from the sixth storm of the Atlantic hurricane season were clocked at 85 kilometres (50 miles) per hour, the Miami-based US National Hurricane Center said.
But, the center said that "Fay could be approaching hurricane strength when it nears the Florida Keys Monday night."
At 0600 GMT, the storm was located about 180 kilometres (110 miles) southeast of Havana.
Fay was moving westwards at 20 kilometres (13 miles) per hour, and expected to turn north later Monday.
"There exists a possibility of moderate flooding on the southern and northern shores of the island and in its western parts," warned Jose Rubiera, director of the Center of Forecasts at the Cuban Institute of Meteorology.
After crossing Cuba, Fay is then expected to head up the west coast of Florida, hitting land near Tampa on Tuesday afternoon and moving straight north into Georgia.
In the Dominican Republic, more than 2,000 people were evacuated to shelters as the storm felled trees, damaged hundreds of houses and uprooted power lines disrupting supplies to more than 15,000 homes, according to local news reports.
At least four people were killed in its deadly trail.
Fay had earlier raked across Haiti, which shares the Caribbean island of Hispaniola with the Dominican Republic, and red alerts were posted along with orders to evacuate flood zones as heavy rains and winds battered much of the country.
"Most of Haiti is under heavy rains, especially in the south, with winds clocked at about 70 kilometres per hour (43 mph)," Ronald Semelfort, an official with Haiti's meteorological service, told AFP.
Civil Protection officials said the heavy rains killed seven people across the country and left three others missing, two near the capital Port-au-Prince and one in the southern part of the country.
Haiti's national weather service lifted a tropical storm warning on the island, but cautioned people living in flood and landslide-prone areas.
With the storm expected to gather strength over the open water between Haiti and Cuba, Havana warned residents in the south to brace for the storm.
More than 15,000 people have been evacuated from Cuba's southern coastal areas between Matanzas and Havana, where Fay is expected to pass.
A hurricane watch was in effect in eight of Cuba's 14 provinces, including Havana. Tropical storm warnings were in force in the Cayman Islands and southwestern Haiti, and a tropical storm watch in central Bahamas and Jamaica.
In Miami, residents descended on gas stations and supermarkets to fill up on petrol and bottled water as well as other emergency items.
Local television stations broadcast warnings about possible business shutdowns and power cuts.