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.
Beijing said it firmly opposed the new tariffs but China’s ministry of commerce said it was “confident and capable of meeting any challenge”.
Trump: Tariffs on $60bn in Chinese goods
US exempts EU from steel tariffs
Reality Check: Is Trump right about US trade deficit?
Trump slaps big tariffs on metal imports
“China will not sit idly by its own legitimate rights and interests. We are fully prepared to defend our legitimate interests,” the ministry said.
But it said it hoped the US would not drag bilateral economic and trade relations into danger.
Fears of a trade war pushed Asian stock markets down sharply. China’s Shanghai Composite Index closed down 3.4% while Hong Kong’s Hang Seng index ended 2.5% lower.
However, the falls were less severe in Europe. Germany’s Dax index dropped 1.7%, with Volkswagen and steel maker ThyssenKrupp among the biggest losers.
The UK’s FTSE 100 share index was 0.8% lower.
On Thursday, US stocks had tumbled, with the S&P 500 index ending the day down 2.5%.