The newspaper reported that Cook travelled to the White House last month to warn Trump of the potentially adverse effects of Trump's trade policies on Apple in China but did not specify precisely when Trump made the commitment to Cook.
Neither side has yet imposed tariffs on the other in their growing dispute over technology and the US trade gap; the first round is to take effect on July 6.
Louis Kuijs, head of Asia Economics at Oxford Economics, agreed that China would not be able to compete with the U.S. on a tariff-per-dollar of imports basis.
President Trump said the tariffs on these goods would be 10 percent. "Rather than altering those practices, it is now threatening United States companies, workers, and farmers who have done nothing wrong", he said.
Global stock markets tumbled with US bond yields and agricultural commodities on Tuesday, while the dollar rose and investors flocked to safety in the face of a rapidly escalating U.S.
European Union countries last Thursday approved a raft of tit-for-tat tariffs on up to 2.8 billion euros ($3.2 billion) worth of U.S. goods, including emblematic exports like jeans, whiskey and motorcycles. All told, Trump is now threatening to penalize up to $450 billion of Chinese goods - a value representing about 90 per cent of Chinese imports a year ago.
China's commerce ministry condemned the announcement, saying the U.S. was engaging in a "practice of extreme pressure and blackmail".
China will protect its interests will take qualitative, quantitative measures if us publishes additional tariffs list.
The move, which drew a strong response from Beijing, follows China's announcement that it would retaliate in kind to Trump's initial decision last week to hit $50 billion worth of Chinese goods with a 25 percent tariff.
The U.S. blocked ZTE's access to U.S. suppliers in April, saying the company violated a 2017 sanctions settlement related to trading with Iran and North Korea and then lied about the violations.
The tit-for-tat moves could start to meaningfully slow US growth, economists warn.
The Australian dollar sank to a one-year low of A$ 0.7391 as the U.S.
The tariffs could affect $50-$60 billion worth of goods and increase trade tensions. It fell 99 cents to $64.86a barrel in NY, and Brent crude, the worldwide standard for oil prices, fell 0.7 per cent to $74.78 a barrel in London.
Tariffs on other USA goods will be announced at a later date, the Xinhua news agency reported. He added that China is a "predatory economic government" that is "long overdue in being tackled", matters that include IP theft and Chinese steel and aluminum flooding the USA market.
The US president said the tariffs would be imposed if China "refuses to change its practices". That would encompass roughly 90 percent of the $505 billion worth of goods that China exported to the United States in 2017. That means things like tourism and education, industries from which the United States benefits a lot more than China does.