The US National Hurricane Center said Claudette, packing winds of nearly 110 kilometers (70 miles) an hour with higher gusts, was expected to reach Mexico's Yucatan peninsula late Thursday or early Friday.
"Claudette has become less organized ... However, there is still the potential for it to become a hurricane as it approaches the Yucatan peninsula."
A tropical storm warning remained in effect along the Yucatan from Progreso to Campeche and along the Belize coast from Belize City northward to the Mexican border.
Tropical storm warnings for Grand Cayman island were to be downgraded late Thursday, the center said.
At 2100 GMT, Claudette was centered about 340 kilometers (210 miles) southeast of Cozumel, Mexico, the northeastern tip of the Yucatan. It was advancing northwest at 22 kilometers (14 miles) per hour.
US authorities said the Mexican authorities in Yucatan had issued a hurricane warning along the threatened coast from Chetumal to Catoche.
After crossing the Yucatan Peninsula, Claudette will cross the Gulf of Mexico and head for Texas, with landfall expected early next week, the hurricane center said.
The Atlantic hurricane season runs from June to November. Experts have predicted more storms than usual this year.