Forecasters said the storm's likely path would take it away from the site of the huge Gulf of Mexico oil spill off Louisiana's coast, but added that it might push oil farther inland and disrupt cleanup efforts.
Alex had maximum sustained winds near 65 mph (100 kph) early Tuesday, and the U.S. National Hurricane Center in Miami, Florida, predicted the storm would grow into a hurricane sometime Tuesday as it headed toward the U.S.-Mexico border at the mouth of the Rio Grande. Landfall seemed likely Wednesday night.
Forecaster Todd Kimberlain said conditions Monday led the center to conclude the storm would be a less powerful hurricane than initially thought.
Tropical storm-force winds extended up to 80 miles (130 kilometers) from the storm's center, and Alex was moving toward the north-northwest at 8 mph (13 kph).
A hurricane warning was posted for the Texas coast from Baffin Bay, 100 miles (160 kilometers) south to the mouth of the Rio Grande river and on a further 225 miles (360 kilometers) to La Cruz, Mexico. Except for the border area itself, both regions are lightly populated.
Workers along the South Texas coast were clearing drainage ditches, filling sandbags and positioning heavy equipment and water pumps as well as preparing emergency shelters. Some cities also handed out sandbags to residents and urged people to make preparations.
Mexico's northern Gulf coast braced for heavy rains like those that fell on southern areas and parts of Central America earlier.
"It is a fact we are going to get very heavy rains," said Gov. Fidel Herrera of the Gulf coast state of Veracruz.
Forecasters said rain from Alex would keep falling on southern Mexico and Guatemala into Tuesday, raising the possibility of life-threatening floods and mudslides
The hurricane center estimated that Alex would dump five to 10 inches (12.5 to 25 centimeters) or rain over portions of northeastern Meixco and southern Texas over the next few days.
Heavy rains in Mexico's southern Gulf coast state of Tabasco forced the evacuation of about 300 families from communities near the Usumacinta river.
Alex caused flooding and mudslides that caused at least five deaths in Central America over the weekend, though Belize and Mexico's Yucatan Peninsula appeared largely unscathed.
The storm made landfall in Belize on Saturday night as a tropical storm and weakened into a depression on Sunday as it crossed the Yucatan Peninsula.
When Alex became the first named storm of the Atlantic hurricane season, officials immediately worried what effect it could have on efforts to contain the millions of gallons (liters) of crude spewing into the northeastern part of the Gulf.
A cap has been placed over the blown-out undersea well, directing some of the oil to a surface ship where it is being collected or burned. Other ships are drilling two relief wells, projected to be done by August, which are considered the best hope to stop the leak.
Stacy Stewart, senior hurricane specialist at the U.S. hurricane center, said early Monday that Alex's center wasn't expected to approach the oil spill site, but the storm's outer wind field could push more oil onto land and hinder operations in the area.
Alex was centered about 475 miles (765 kilometers) southeast of Brownsville, Texas, early Tuesday. Its rains could reach Veracruz and the border state of Tamaulipas late Tuesday or Wednesday, the hurricane center said.