Friends and relatives of the passengers said they were receiving messages indicating uncomfortable conditions on the ship, which is stranded in the Gulf of Mexico, 150 miles off the coast of the Yucatan peninsula, after an engine room fire.
An official of Carnival Lines said 4,200 people are on board.
"We have no power at all, which means we can't use the toilets, wash our hands or take a shower," said a text posted on the CruiseCritic.com website.
Relatives of passengers said messages received include complaints about sleeping on deck and shortages of food and water, USA Today reported Monday.
A passenger in need of dialysis was to be removed from the ship to another Carnival vessel later Monday,Petty Officer 3rd class Richard Brahm, a U.S. Coast Guard spokesman, said.
There were no injuries or deaths among the guests or crew as a result of the Sunday fire, KLTV-TV, Tyler, Texas, reported.
The ship's automatic fire extinguishing system kicked in and contained the blaze, CNN reported. The cause of the fire was not reported.
The Carnival Triumph was en route to its home base of Galveston, Texas, though it will now be towed to Mexico, and passengers and crew will be flown back to the United States, said Astevia Gonzalez of the Carnival Cruises family support team.
"A tugboat is en route to the ship's location and will tow the vessel to Progreso, Mexico, which is the closest port," Carnival said in a statement.
The vessel should arrive at the port by Wednesday, Gonzalez said.
Guests on the Carnival Triumph will get a full refund, a credit to be used toward a future cruise and reimbursement for all expenses, CNN reported.