Carnival hasn't determined what caused the fire, said Oliva, the company spokeswoman.
The National Transportation Safety Board announced Tuesday it has opened an investigation into the cause of the fire. The NTSB said the Bahamas Maritime Agency will lead the investigation because the ship carries a Bahamian flag.
The ship was originally going to be towed to a port in Progreso, Mexico, but after currents pushed it northward, a decision was made to take it to Alabama to make it easier for passengers without passports to get home, the company said.
Cahill said Carnival has reserved more than 1,500 hotel rooms in Mobile and New Orleans for Thursday. The company plans to return passengers back to Houston Friday using charter flights.
A similar situation occurred on a Carnival cruise ship in November 2010. That vessel, named Splendor, was stranded with 4,500 people aboard after a fire in the engine room. When the passengers disembarked in San Diego, they described a nightmarish three days in the Pacific with limited food, power and bathroom access.
Cahill said the Spendor's fire was different because it involved a "catastrophic explosion" in a diesel generator, and the Triumph's fire had "some other cause." He could not say what the economic impact will be due to the fire aboard the Triumph. The impact from the Splendor was $40 million, he said.
Carnival canceled the Triumph's next two voyages, scheduled to depart Monday and Saturday. Passengers aboard the stranded ship will also receive a full refund.