The reported attacks began last Wednesday as people in Hong Kong, a former British territory, prepared to vote in a pro-democracy referendum, it said.
"At its peak, the distributed denial-of-service attack launched on June 18 flooded Apple Daily in Taiwan with more than 10 billion bogus DNS requests in a 24-hour period," Next Media said. "Web logs from a separate, direct attack on Apple Daily servers show much of the traffic originated from IP addresses in China, though some traffic came from Russia and the United States."
The determination by the company came after viewing internal and vendor records.
"The cyber-attacks came as the people of Hong Kong prepared to launch a referendum on how to negotiate Hong Kong's democratic future with China," it said. "Apple Daily and Next Media Group believe that all people should be able to participate in free elections and have openly supported the referendum."
Next Media said China had denounced the referendum as illegal.
The Apple Daily website in Hong Kong has recovered from the cyber-attacks and is operating at a 90 percent level but the site in Taiwan is said to remain hampered by attacks, which are still occurring.