The storm has strengthened in the Gulf of Mexico and is threatening to drop heavy rains on Florida.
"Alberto has the potential to become a hurricane within the next 24 hours," US forecasters in Miami said.
Alberto's maximum sustained winds had increased to nearly 110km per hour (70 miles per hour) and further strengthening was possible, the US National Hurricane Centre said. Tropical storms become hurricanes once their maximum sustained winds reach 119 kph (74 mph).
The hurricane centre said the storm, moving north-northeast at about 11 kph (7 mph), was expected to hold to the same general path over the next 24 hours.
"This is a serious storm and we are taking it seriously," Jeb Bush, the governor of Florida, told reporters in Tallahassee as the state government declared a state of emergency.
Alberto brought heavy rain to Cuba and is expected to reach land north of Florida's heavily populated Tampa-St. Petersburg area on Tuesday, cross the state and enter the Atlantic Ocean.
Rain has started to fall on Florida's west coast and forecasters said 10-20 centimetres of rain were possible across parts of Florida and Georgia.
Florida has been hit by eight hurricanes in the past two years, including Katrina, which went on to devastate New Orleans. More than 1,300 people were killed by Katrina and much of the US Gulf coast is still recovering.
Alberto formed off Cuba, and civil defence officials there reported that 26,000 people evacuated low-lying areas in the Caribbean island's westernmost province of Pinar del Rio, where 40-50 centimetres of rain fell in 24 hours.