The White House did not immediately say what authority the President had to compel
private firms to quit a country, the BBC reported on Friday.
Trump announced a 5 per cent increase on tariffs on Chinese imports, after China unveiled plans for duties of 10 per cent on $75 billion of US goods.
The latest salvos in the trade war sent global financial markets tumbling.
The Dow Jones Industrial Average lost more than 620 points, or 2.4 per cent, while London's FTSE 100 and the German DAX also turned negative.
Stock markets in the US fell following news of China's tariffs, but then recovered, only to fall again in response to Trump's tweets.
The Dow Jones Industrial Average closed down 2.4 per cent at 25,629. The S&P 500 dropped 2.6 per cent while the Nasdaq was 3 per cent lower.
"In the spirit of achieving Fair Trade, we must Balance this very unfair Trading Relationship," Trump tweeted.
"Our great American companies are hereby ordered to immediately start looking for an alternative to China," he tweeted earlier on Friday.
China's new tariffs will range between 5 per cent and 10 per cent and apply to more than 5,000 goods coming from the US.
Agricultural goods, crude oil and small aircraft are among the targeted items.
Trump hit back, tweeting that $ 250 billion of Chinese imports, currently taxed at 25 per cent, would from now on be taxed at 30 per cent.
Beijing also said it would revive a 25 per cent tariff on US car imports that it lifted earlier in 2019 in a goodwill gesture as the two countries tried to negotiate a trade agreement.
Carmakers warned that the tax would put US jobs at risk.
"When these tariffs were initially imposed by China in 2017, American exports of finished vehicles dropped by 50 per cent," said John Bozzella, who represents car manufacturers.
"We can't let that happen to American workers again."
On August 1, Trump unveiled a further 10 per cent tariff on $300 billion of Chinese goods, blaming China for not following through on promises to buy more American agricultural products.
That tariff was expected to be introduced on September 1, but less than two weeks later Trump delayed that date to December 15, partly due to concerns it might hit Christmas shoppers.
China said it planned to impose its new tariffs in two stages on September 1 and December 15.
Trump also turned his fire on Friday against the head of the US central bank after he spoke out about the economic risks of a trade war with China.
The president questioned whether Federal Reserve chairman Jerome Powell was a greater "enemy" than China's leader Xi Jinping.
At a symposium of central bankers in Jackson Hole, Wyoming, on Friday Powell warned that trade tensions were hitting the global economy.
He also said the Fed - the world's most powerful central bank - didn't have a "rulebook" to deal with the fallout.
The White House moved to play down the new tariffs. Trump's trade adviser, Peter Navarro, told CNN that the Chinese duties were "well signalled," adding: "This isn't breaking news."
He said the economic slowdown was the fault of the US Federal Reserve, which has drawn criticism from Trump for not making bigger cuts in interest rates.
Trump also said on Twitter on Friday that the economy was "strong and good", while "the rest of the world is not doing so well".
( With inputs from IANS )