The United States National Hurricane Centre (NHC) say Hurricane Lee is expected to intensify into an “extremely dangerous” category four storm by the weekend.
The Atlantic storm became a category one hurricane, packing 75mph (120 km/h) winds on Wednesday afternoon.
The NHC said it was too early to determine whether it might make landfall in the US.
But swells generated by the storm could reach the Virgin Islands and Puerto Rico by this weekend.
Lee is expected to rapidly intensify into an extremely dangerous major hurricane by Saturday morning.
For now, forecasters have warned the Leeward Islands, where the Caribbean and Atlantic join, to be on alert.
The winds from Lee are projected to reach 145mph (233 km/h) later in the week, which would make it a category-four hurricane.
It comes just days after Category 3 Hurricane Idalia made landfall in Florida, wreaking an estimated billions of dollars in damage, before sweeping over Georgia and the Carolinas.
Major power outages lasted for days in Georgia.
Mid-September is considered to be the peak of the hurricane season, which ends on November 30.
(Editor: Ifeanyi Mark)