Mr Trump lashed out as as senior White House figures accused China of trying to destabilise USA democracy and presenting the most serious threat to its security.
The United States and China have become increasingly embroiled in a trade dispute that has seen the two sides impose tariffs on billions of dollars' worth of goods.
September exports rose 14.5 percent from a year earlier, Chinese customs data showed.
Beijing's export data has been surprisingly resilient to tariffs, possibly because companies ramped up shipments before broader and stiffer US duties went into effect, raising concerns about a sharper drop in export strength once all tariffs kick in.
IMF Managing Director Christine Lagarde warned on Thursday against adding currency wars to the trade conflict, saying this would hurt global growth and "innocent bystander" countries. -China trade frictions "have been escalating and other unstable factors still exist caused by a number of economic uncertainties worldwide", he said.
President Trump has taken a hard line stance against China, vowing to increase tariffs until a fair trade agreement can be reached.
Overall, China's global exports rose 14.5 percent over a year earlier, up from August's 12.2 percent growth.
The U.S. Treasury Department's staff has advised Secretary Steven Mnuchin that China isn't manipulating the yuan as the Trump administration prepares to issue a closely watched report on foreign currencies, according to two people familiar with the matter.
Kudlow said that so far, the administration viewed China's negotiating offers as "rather unsatisfactory" but that "maybe talks between the two heads of state will bear fruit".
Analysts said last month's strong export growth - which might indicate USA tariffs are not biting much yet - is unlikely to be sustained.
Plus, the 10 percent tariffs were nearly entirely offset by the fall in the Chinese currency, which has depreciated by more than 8 percent since June.
"It's obvious that the immediate effects of the trade war are the exact opposite of what the Trump administration had been planning", Andrew Polk of Beijing-based research firm Trivium China tells the Post.
"He won't be happy with these figures but it proves that tariffs don't help curb exports", he said.
Trump is pressing China to improve trading conditions for USA products and to end what U.S. businesses say is widespread theft of their intellectual property. "Americans are not stupid people", Trump said in an interview on "Fox and Friends". "Their economy has gone down very substantially and I have a lot more to do if I want to do it".
The state-owned China General Nuclear Power Company (CGNPC) was also charged.
China is not backing down as tensions escalate with the United States, issuing a mock-proposal yesterday that the trade deficit would be easily closed if America sold expensive military equipment to Beijing. But sellers of factory machinery and other more advanced exports express confidence they can keep their market share.