Mr. Trump’s sweeping tariffs on steel and aluminum imports and on $34 billion of Chinese goods have thrown the global trading system into a state of upheaval. Retaliation from the European Union and China is starting to affect American businesses and consumers, and the White House is considering whether to levy a new round of duties on car imports.

Brushing off cries of protectionism, Mr. Mnuchin was preparing to press for a new trade deal between the United States and the European Union that would eliminate all tariffs and non-tariff barriers to trade. The idea, which Mr. Trump has broached before, has emerged as America’s latest strategy to repel accusations that it has abandoned free trade.

“I think the G-7 and the E.U. constantly talk about a rules-based system, eliminating tariffs and trade barriers and how good free trade is for the global economy,” Mr. Mnuchin said in an interview before his meetings. “If they’re sincere, we’re willing to sit down and negotiate those discussions.”

The proposal is largely conceptual at this stage, and it is not entirely clear how it would work. Mr. Mnuchin suggests that the pact would substantially reduce America’s trade deficit with the European Union as a percentage of gross domestic product. However, both sides are likely to seek exceptions and loopholes to protect vulnerable industries such as steel and automobiles in the United States.

Next week, European Union officials, including Jean-Claude Juncker, the president of the European Commission, are to hold additional trade talks with Mr. Trump in Washington. The officials’ visit is an attempt to stave off new tariffs on European car imports.

The United States will have about a dozen bilateral meetings this weekend and will participate in a broader meeting of the Group of 7 countries, which is intended to focus on China’s trade practices. Mr. Mnuchin has no one-on-one meetings scheduled with Chinese officials, but he said that he expected informal talks to take place. (Liu He, China’s vice premier and the country’s top economic official, is not in attendance.)

Thus far, the trade dispute with Beijing shows no signs of abating. Mr. Trump said in an interview with CNBC that aired on Friday that he was prepared to impose tariffs on $500 billion worth of Chinese imports — approximately all the goods China sends to the United States each year — if necessary.