Pulled by tugboats, the Carnival Triumph is due later to reach Mobile, Alabama, although it is still hours from shore.
Buses will be waiting for the vessel’s passengers when they arrive.
Passengers have reported sewage on the floors, poor sanitation and access to toilets, and lengthy queues for food.
Carnival Cruise Lines, which operates the ship, says the Triumph’s 3,000 passengers would have the option of boarding buses to New Orleans or going to Texas.
Galveston, Texas, from where the cruise embarked last Thursday, is seven hours from Mobile. New Orleans, where the firm said it booked 1,500 hotel rooms, is two hours away.
Those staying in New Orleans will be flown on Friday to Houston, Texas.
The powerless vessel is expected to dock between 20:00 and 23:00 local time on Tuesday.
A former cruise terminal was being preparing to help passengers go through customs.
Carnival employees, of whom there are more than 1,000 aboard the ship, will be staying in Mobile.
Mayor Sam Jones said the city had more than enough hotel rooms, as well as two airports near the cruise terminal.
Passenger Janie Baker told NBC by phone on Thursday that conditions on the ship were “extremely terrible”. There has been no electricity and few working toilets, she said.
Janie Baker described using plastic bags to go to the bathroom and that she had seen a woman pass out while waiting for food.
“It’s just a nightmare,” she said.
Janie Baker said she and her friends slept with their life vests one night because the ship was listing.
Passengers are supposed to get a full refund and discounts on future cruises. Carnival announced on Wednesday they would each get an additional $500 in compensation.
But the firm has disputed the accounts of passengers who describe the ship as filthy, saying employees are doing everything to ensure people are comfortable.
A spokesman, Vance Gulliksen, said the company was choosing to bus passengers to New Orleans because it “offered additional capacity and flexibility, which was important to us given that the complexity of the towing operation creates an uncertain arrival time in Mobile”.
Carnival has cancelled more than a dozen planned voyages aboard the Triumph, while acknowledging that the crippled ship had other mechanical problems in the weeks before the fire.
Vance Gulliksen said Triumph had an earlier electrical problem with the ship’s alternator but that repairs were completed by February 2. He said there was no evidence linking the previous problem to Sunday’s fire.
The National Transportation Safety Board has opened an investigation into the fire.
No-one was injured in the blaze, but one passenger with a pre-existing condition was taken off the ship as a precaution.
“We know it has been a longer journey back than we anticipated at the beginning of the week under very challenging circumstances,” Carnival president and chief executive Gary Cahill said in a statement.
“We are very sorry for what our guests have had to endure.”
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