China claims self-ruled Taiwan as part of its territory and relations between the two have deteriorated in recent years.
A series of fishing boat incidents occurring along the narrow waterway separating Taiwan and China have heightened tensions.
The latest incident occurred early on Saturday when the Chinese-flagged boat "Min Long Yu 60877" sank after crashing into an unidentified vessel about 6.5 nautical miles off the coast of the Kinmen islands, Taiwan's Coast Guard Administration said in a statement.
"There were seven crew members on board. Four were rescued and three were missing," it said.
It said a patrol boat sent to the area could not find the missing crew.
"Those who fell into the sea were not found."
The statement said Taiwan's coast guard and its Chinese counterparts were carrying out "an expanded search and rescue" in nearby waters.
Kinmen county is administered by Taiwan but is located just five kilometres (three miles) from the Chinese coastal city of Xiamen.
A fatal incident involving a Chinese boat near Kinmen on February 14 kicked off a months-long row between Taiwan and China.
A boat capsized while it was being pursued by Taiwan's coast guard, killing two Chinese crew members, for which Beijing blamed Taipei.
The two sides reached an agreement in July after negotiations over the incident, agreeing that the cause of death was "drowning".
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