As of 02:00 GMT, key Japanese index Nikkei 225, which reflects the performance of 225 leading Japanese companies, was down 2.33 percent from Friday to 20,228.50.
By the same time, Hang Seng, the main index on the Hong Kong Stock Exchange, fell 3.44 percent to 25,280.
On Friday, US President Donald Trump said that Washington would raise the existing tariffs on $250 billion worth of Chinese goods from 25 percent to 30 percent starting 1 October. Additionally, tariffs would be raised from 10 percent to 15 percent on another $300 billion worth of Chinese goods starting from 1 September.
Earlier the same day, the Chinese State Council said tariffs ranging from 5 to 10 percent would be applied to $75 billion worth of US goods, with some taking effect on 1 September and others on 15 December. The latter batch will also include a 25-percent duty on US cars.