US-based car reviewer Matt Farah said on Twitter that a Porsche he had ordered was on the ship.
“I just got the call from my dealer,” he tweeted. “My car is now adrift, possibly on fire, in the middle of the ocean.”
The fire was still reportedly active as of Thursday night and the cause is not yet known. The ships are often divided into sections, which could limit the damage to the cargo.
The ship is operated by Mitsui OSK Lines, which said it “regrets the inconvenience and concern caused”.
It added: “MOL will make every effort to contain the damage and resolve the situation as the main priorities.”
The crew are safe in a hotel on the Azores and are in good health, it said, adding that it had appointed a salvage team which is due to arrive at the vessel on Friday.
Among the 3,965 cars affected were Volkswagen and Lamborghinis, according to the German newspaper Handelsblatt.
Container ship fires are rare, but can be catastrophic. In 2018, a fire on the Maersk Honam claimed the lives of five of its 27 crew after chemicals it was carrying caught fire and the ship’s carbon dioxide fire suppression system was unable to bring the blaze under control. It was salvaged and renamed the Maersk Halifax.
Container ship sizes have increased by 1,500pc since 1968, according to insurer Allianz, which has warned that ever-more sophisticated fire-fighting systems will be needed to keep up.
Volkswagen did not immediately respond to a request for comment.