Emergency services are working to free people who are trapped in a hotel that has partially collapsed, the reports say.
Efforts to rescue people trapped inside is underway.
No tsunami warning has been issued.
Two people were killed and more than 100 injured in a 6.5-magnitude natural disaster which jolted waters near Taiwan's Hualien County at 11.50 p.m. on Tuesday.
Scenes of widespread structural damage in the streets have been shown on numerous Facebook live videos filmed by locals.
"Straightaway we saw the roads with all the cracks". The quake, which was one of the deadliest in the world had struck about 30 km south of the Iraqi town Darbandikhan, near the north-eastern border with Iran.
Margaret K. Lewis, a Seton Hall University Law School professor living in Taipei, said she felt prolonged swaying at her modern high-rise apartment building in Beitou District, in the northern part of the city. "I wouldn't be surprised for there to be a number of aftershocks in the next couple of days". We dare not go back home now.
The Hualien Shangxiao Road Community Apartment suffered severe damage and was tilted 45 degrees, the first and second floors nearly completely crushed, Taiwan News reported.
The Ministry of Foreign Affairs and Trade said it had received no requests for consular assistance following the quake.
"At the time of publication, Taiwan had registered 19 earthquakes today, with all the tremors clustered around the same area off the coast of Hualien".
A magnitude 6.1 natural disaster struck nearby on Sunday.