Japan's Nikkei share index fell 4.5% and in the US the Dow Jones sank 2.9%.
China responded to news of the planned tariffs by saying that while it did not want a trade war, it was "absolutely not afraid" of one.
Mr Trump's proposed tariffs are a response to allegations of intellectual property theft by China.
The US launched a complaint against China at the World Trade Organization (WTO) on Friday.
Its statement said: "China appears to be breaking WTO rules by denying foreign patent holders, including US companies, basic patent rights to stop a Chinese entity from using the technology after a licensing contract ends."
"China also appears to be breaking WTO rules by imposing mandatory adverse contract terms that discriminate against and are less favourable for imported foreign technology."
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