FRANKFURT — President Trump’s threat to impose tariffs on imported cars is working, sending foreign leaders from Mexico to Japan racing to the negotiating table to make deals. But their offers may not satisfy a president who has shown a willingness to embrace trade tensions in an effort to extract more from American trading partners.

The European Union on Thursday made one of the most significant concessions, saying it would cut its existing penalties on automobiles to zero, provided the United States dropped its own tariffs. Although the president has called for something similar, Mr. Trump said he wanted Europe to go even further.

“It’s not good enough,” Mr. Trump said in an interview with Bloomberg News. “Their consumer habits are to buy their cars, not to buy our cars.”

If deals do not get done, the president’s combative approach to trade is risky, leaving American companies vulnerable to both tariffs in the United States and retaliatory measures by other countries. Car manufacturers like General Motors and Ford depend heavily on a global supply chain. Tariffs could raise sticker prices and cost jobs.