At around 5 p.m., Dolly's centre was about 165 miles east-southeast of Brownsville, Texas, moving northwest at about 10 miles per hour.
In Tamaulipas, on the U.S. Mexico border, residents were seen preparing for the hurricane by filling up sandbags and stocking up on supplies at local supermarkets.
Dolly was expected to make landfall late Tuesday or early Wednesday as a Category 1 hurricane and bring with it high winds and up to 15 inches of rain and coastal storm surge flooding of four feet to six feet above normal high tide levels.
Emergency officials feared major flooding problems and urged coastal residents to prepare.
Mexico's Minister of the Interior Juan Camilo Mourino made a trip to the border state of Tamaulipas and assured residents that the state and federal arms were prepared for the storm.
"The state, with its systems and its Civil Protection unit, is ready to deal successfully with the challenge we are facing," he said at a news conference with the governor of Tamaulipas.
Mexican border towns near the Gulf coast were setting up shelters for those who want to leave low-lying areas that flood easily.
"We're evacuating around 23,000 people to safer areas, the people who have a higher level of risk in terms of the route that the tropical storm looks like it's going to take," Mourino said.
Soldiers were also being sent into Matamoros to protect against looting, in case the storm does strike the Mexican border town on the Gulf Coast and many residents are forced to flee.
Mexico announced a hurricane warning from Rio San Fernando north to the U.S. border.
A tropical storm warning and a hurricane watch were also in effect from La Pesca to Rio San Fernando.
Tropical storm warnings were issued for areas adjacent to the hurricane zone, and the governor declared 14 South Texas counties disasters, allowing state resources to be used to send equipment and emergency workers to areas in the storm's path.
Officials are concerned about levees breaking.
Much of the damage to New Orleans from Hurricane Katrina was from levee breaks instead of wind.
Shell Oil said it was evacuating workers from oil rigs in the western Gulf of Mexico, but didn't expect its production to be affected by the storm.
