Packing maximum wind speeds of 155 kilometres (96 miles) per hour, the storm could make landfall late Wednesday or early Thursday, the Central Weather Administration said.
"It is now intensifying rapidly," Lin Po-tung of the weather agency told a news conference.
Lin warned waves could reach five to six metres (16 to 20 feet) high on Wednesday, with heavy rain also forecast in the capital Taipei.
Disaster officials in Taitung county, which looked set to bear the brunt of Kong-rey based on the storm's current trajectory, advised fishers to return to shore and secure their boats, while ferry services to outlying islands were suspended.
"The main impact on Taitung will be damage caused by strong winds," the local fire agency said.
Further north in the coastal city of Hualien, which was hit by a magnitude 7.4 earthquake in April, authorities prepared supplies for the vulnerable and vehicles ready to evacuate people.
Taiwan is accustomed to frequent tropical storms from July to October but scientists have warned climate change is increasing their intensity, leading to heavy rains, flash floods and strong gusts.
In July, Gaemi became the strongest typhoon to make landfall in Taiwan in eight years, killing at least 10 people, injuring hundreds and triggering widespread flooding in the southern seaport city of Kaohsiung.
That was followed in early October by Krathon, which killed at least four people and injured hundreds, as well as triggering mudslides, flooding and record-strong gusts.
In a bid to avoid a repeat of the flooding, there have been "increased efforts on clearing sediment from rivers and in more areas", said Yi-fung Wang, a spokesman for the Water Resources Agency, under the economic ministry.
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