Stormy weather in the wake of Typhoon Gaemi has prevented rescuers from reaching the sailors, Hsiao Huan-chang, head of Taiwan's National Fire Agency, told a briefing.
"They fell into the sea and were floating there," Hsiao said, adding that rescuers contacted a nearby Taiwanese cargo ship to assist them.
Hsiao did not specify when the Tanzania-flagged ship sank, but said the rescue vessel arrived in the area at 8:35 am (0035 GMT).
"(When the Taiwanese ship arrived) the visibility at the scene was very low and the winds were too strong," he said.
"When the weather permits, we will immediately dispatch ships or helicopters to rescue but at the moment it is not possible."
Another official at the agency told AFP after the briefing that the sailors were missing.
News of the ship's sinking came after Typhoon Gaemi made landfall in Taiwan on Wednesday night with sustained wind speeds of 190 kilometres (118 miles) per hour at its peak.
Authorities said it was the strongest typhoon to hit Taiwan in eight years.
At least two people in Taiwan were killed due to Gaemi's impacts, including a woman who was crushed after a part of a building fell on her.
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