Heavy rainfall and possible isolated flash flooding and mudslides will likely hit Barbados and the Windward and Leeland Islands by Friday, forecasters said.
Potential impacts on Florida weren't immediately clear "given the uncertainty in the long-range forecast," the hurricane center said.
Parts of the Virgin Islands, Puerto Rico, Cuba, Hispaniola, Turks and Caicos and the southeastern Bahamas will likely have wind and rain through early next week, the NHC said.
The tropical storm was moving west with current maximum sustained winds of 40 mph.
A faster west-northwesterly movement was expected within the next 24 to 36 hours, the hurricane center said.
A Tropical Storm Warning is in effect for:
- St. Lucia
- St. Vincent and the Grenadines
A Tropical Storm Watch is in effect for:
Tropical Storm Danny weakened to a tropical depression hours after it made landfall Monday evening on South Carolina's coast but the system continued to dump heavy rains on the state and portions of Georgia.
The Associated Press contributed to this report.