The "Hsiang Fu Chun", a 700-tonne squid fishing vessel, lost contact with its owners "soon" after reporting that water was leaking on to the deck at around 3:00 am on February 26, officials said.
The vessel was sailing about 1,700 nautical miles (3,148 kilometres) off the Falkland Islands when it vanished, according to recorded satellite data.
Its crew include a Taiwanese skipper and chief engineer, as well as 11 Chinese, 21 Indonesian, 13 Filipino and two Vietnamese sailors.
"We still don't know where the ship is and what happened to it," Huang Hong-yen, spokesman for the Fisheries Agency, told AFP, adding that the government had launched a search-and-rescue effort "immediately" after the ship's owners said it had lost contact.
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He said there was no evidence the boat had sunk. The ship was equipped with a system that automatically issues a mayday signal when placed under a certain water pressure, but no such signal was sent, he added.
Some Taiwanese media have speculated that the vessel could have lost power and be adrift, or could have been hijacked by crew.
But another official from the agency who asked not to be named said: "To be honest, the hope of finding the ship in that remote area is fading."
The South Atlantic Ocean is a traditional fishing ground for Taiwanese vessels, attracting up to 100 squid boats from the island each year.