The Florida-based National Hurricane Centre (NHC) has said the storm could produce isolated tornadoes in southern Texas.
Many residents in the area have been preparing for the storm for days, readying their homes and businesses and stocking up on household essentials.
Barack Obama, the US president, has declared a state of emergency in Texas clearing the way for federal aid for relief operations.
Alex is the first Atlantic hurricane to form this early, in June, since 1995, according to the NHC.
The storm has already killed at least 10 people in Nicaragua, Guatemala and El Salvador over the weekend.
The hurricane is expected to stay clear of the main oil fields in the Gulf of Mexico and areas worst hit by the massive BP oil spill, but strong winds have already caused problems for the clean-up effort there.
|The storm has forced the suspension of oil spill clean-up efforts in the Gulf of Mexico [AFP]
Many coastal operations have been suspended and boats used to skim oil from the sea surface have been ordered to return to port for safety.
Controlled burns of oil on the surface and flights spraying dispersant chemicals have also been halted.
Meanwhile the winds and heavy seas have threatened to push more of the huge slick onto shore, particularly in the already badly-hit US states of Louisiana, Mississippi, Alabama and Florida.
However the hurricane was not expected to hamper the operation siphoning oil from the ruptured BP well, or the drilling of relief wells which engineers say will eventually bring a halt to the leak.
The ongoing oil spill began in April following an explosion on the BP-operated Deepwater Horizon oil rig, which left 11 workers dead.
Since then, between 1.6m and 3.6m barrels of oil have poured into the Gulf from the ruptured undersea wellhead, fouling large stretches of US coastline, killing wildlife and devastating coastal fishing communities.