The storm was over open sea, centered roughly 370 miles east-southeast of Kingston, Jamaica, while its outer fringes spread rain and scattered thunderstorms across parts of the Caribbean.
The storm was drifting west at 25 mph and could become a hurricane within 48 hours, forecasters at the U.S. National Hurricane Center in Miami said.
"Especially if it reduces its forward speed, it could develop to hurricane strength," said Lt. Dave Roberts, a Navy meteorologist at the center. "Right now it's in a favorable environment, so it just needs to get some time to pull itself together."
The system was upgraded from a disorganized tropical wave Tuesday afternoon after an Air Force Reserve Hurricane Hunter detected sustained winds of 50 mph and a low-level circulation center.
Tropical storm-force winds extended out to 70 miles from the storm's center.
Jamaica issued a tropical storm warning and the Cayman Islands issued a tropical storm watch. Heavy rainfall of up to 6 inches was forecast in the storm's path.
A gale warning was in effect for southern Caribbean waters and small craft advisories were in effect for all coastal waters.
Strong winds and heavy rains associated with the system were expected to spread over Jamaica Tuesday night or Wednesday.
"We're looking at further strengthening over the next 24 hours and possible hurricane strength in 48 hours," Roberts said.
Tropical storms become hurricanes when their sustained winds surpass 74 mph.
Roberts said the storm was projected to continue westward, reaching the Yucatan Peninsula by midday Thursday.
Claudette is the third tropical storm to form this year in the Atlantic basin, including the Caribbean. Experts have said this hurricane season, which began June 1 and ends Nov. 30, will be very busy.
As the weather system passed through the Leeward Islands Monday, government offices and private businesses closed in Barbados. The main commercial district in the capital, Bridgetown, also closed early after the Barbados Meteorological Office issued a severe weather warning. No damage was reported.