A blowout at the well on Tuesday morning forced the evacuation of 44 workers from the platform.
US Coast Guard and federal safety officials are still trying to assess the potential hazards.
The area was hit by the Deepwater Horizon rig that exploded in 2010, leaking millions of gallons of oil.
Eleven oil rig workers were killed in what was the worst US offshore disaster.
The latest blowout was not of that magnitude, officials told the Associated Press news agency.
On Wednesday morning the Bureau of Safety and Environmental Enforcement (BSEE) said the fire was damaging the rig structure.
"As the rig fire continues, the beams supporting the derrick and rig floor have folded and have collapsed over the rig structure," the agency said in a statement.
But after an aerial tour of the rig, no gas sheen was visible on the water surface.
One Coast Guard cutter, Pompano, is near the scene and another, Cypress, is travelling to the area.
In addition, "a third vessel equipped with fire-fighting capability and improved monitoring system is enroute," the BSEE added.
The portable drilling rig - which operates in shallow waters of 154ft (47m) - is owned by Hercules, a contractor for the exploration and production company Walter Oil & Gas Corporation.
The BSEE said the fire broke out while workers were completing construction of a "sidetrack well". The purpose of the sidetrack well was not immediately clear, but industry analysts say they are sometimes used if there is a problem with the main well.
The BSEE said it was investigating the cause of the fire, along with the Coast Guard.
Industry experts are at the scene to try to work out how to bring the well fire under control.
A fire has broken out on a rig drilling for gas in the Gulf of Mexico, 55 miles (85km) off the Louisiana coast, US officials say.