Passengers aboard the stranded Carnival Triumph cruise ship that floated aimlessly in the Gulf of Mexico after an engine fire broke out on Sunday have spoken out on the situation.

Using intermittent phone reception, passengers have told relatives and media outlets of carpets soaked in urine, raw sewage running down the walls, and fights breaking out over the limited food stocks.

The Carnival Triumph. Supplied.

The first tug boat reached the vessel on Monday night and began towing it to Mobile, Alabama at the rate of just a few kilometres per hour.

The ship had been floating off Mexico’s Yucatan Peninsula after a fire erupted in the aft engine room early Sunday, knocking out the ship's propulsion system. Back-up generators are providing limited power.

Some holidaymakers have been left with no alternative but to sleep on the top deck of the vessel, with their lower level cabins completely cut off from fresh air and sanitation.

There are reportedly only five working bathrooms for the 3,000+ passengers on board. The rest are being asked by the crew to urinate in cabin showers and defecate in bags.

Supplies have been ferried to the stricken vessel by nearby ships, including the Triumph’s sister-ship the Carnival Legend. Another ship, the Conquest, delivered additional food supplies.

Passenger Ann Barlow told CNN: "It's disgusting. It's the worst thing ever".

While another passenger, Debra Rightmire, texted ABC: "Toilets are overflowing in the cabins, we are having to sleep in the hallways. Onion and cucumber sandwiches last night".

Carnival spokesperson Vance Gulliksen told AFP that the "overall mood with guests is good under the circumstances and guests are making the most of the situation".

Carnival Triumph is expected to crawl into the Alabama port this Thursday.

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