The storm was expected to make landfall at near hurricane strength as early as Tuesday, said Miles Lawrence, a forecaster with the center in Miami.
Sunday morning, Claudette was about 315 miles east of Brownsville and moving at about 7 mph to the west-northwest. Its maximum sustained winds had increased to 60 mph, 14 mph shy of hurricane strength.
The hurricane watch was issued for the Texas coast from Port O'Connor, about 70 miles northeast of Corpus Christi, to Brownsville, along the Mexican border. Mexican authorities issued a hurricane watch from the border to Rio San Fernando.
The tropical storm swept over the tourist resort of Cancun early Friday, battering high-rise hotels with high winds, flooding several streets and closing the international airport for several hours. But life quickly returned to normal.
Claudette is the third tropical storm of the Atlantic hurricane season. It developed Tuesday in the Caribbean, brushing Jamaica's southern coast with heavy rain and rough surf, battering the Cayman Islands with dangerous waves and above-normal tides and scattering rain over parts of Cuba before arriving in Mexico.
Experts have predicted a busy Atlantic hurricane season, which began June 1 and ends Nov. 30.