The Panama-flagged ship was travelling from Emden, Germany - where Volkswagen has a factory - to Davisville in the United States, based on the Maritime Traffic website.
The 22 crew members on board were evacuated on Wednesday, when the fire broke out, with no one hurt, Portugal's navy said in a statement.
Around 1100 Porsches and 189 Bentleys were on board, spokespeople for the car brands said. Audi, another Volkswagen brand, confirmed some of its vehicles were also on board but did not state how many. It was not clear whether the cars were damaged.
An internal email from Volkswagen USA stated that the ship was carrying 3965 vehicles of the VW, Porsche, Audi and Lamborghini brands, German newspaper Handelsblatt reported.
Volkswagen did not confirm the total number of cars on board and said it was awaiting further information.
YouTuber Matt Farah, whose automotive review channel "The Smoking Tire" has more than a million followers, said on Twitter he was contacted by a car dealer who said the Porsche he ordered was aboard the ship.
"My car is now adrift, possibly on fire, in the middle of the ocean," Farah tweeted.
A picture shared by the Portuguese maritime authority on its website on Wednesday showed clouds of smoke billowing from ship.
The captain of Horta port on the Azorean island of Faial told Portuguese news agency Lusa on Thursday night that the fire was still active but under control.
Azorean authorities were not immediately available for comment.
James Turner, a lawyer specialising in cross-border commercial and shipping disputes, said electrical faults were a common cause of such fires. He said if electric vehicles with lithium-ion batteries were on board, specialist firefighting equipment would be needed to extinguish the flames.
He said vehicle carrier vessels are usually constructed like a multi-story carpark and sealed between each floor, which could limit the damage to the cargo.
