For a president who advertises himself as a paramount deal maker, the next 11 weeks will be a pivotal test, as his advisers race to accomplish what no other administration has done before and reach dozens of individual trade deals with other governments.

President Trump has promised big gains for American trade, and officials from Japan, South Korea, India and elsewhere have been pushing for agreements as they look to forestall punishing tariffs. But trade experts say the administration has set up a seemingly impossible task, given that traditional trade deals typically take months or years to negotiate.

Mr. Trump has tried to use tariffs as leverage to notch quick agreements, and his trade adviser, Peter Navarro, has promised “90 deals in 90 days.” But the levies are creating chaos and financial pain for many businesses, and they have not brought some of America’s largest trading partners, including China, to the table.

Some U.S. trade with China has ground to a halt after the countries imposed triple-digit tariffs on each others’ products, and a wave of bankruptcies, especially among small U.S. businesses that rely on Chinese imports, appears to be looming if the trade barriers are maintained.

Some Trump officials recognize that the situation with China is not sustainable and have been strategizing how to reduce the tariffs between the countries, two people familiar with the discussions said. Another person familiar with the discussions said administration officials were concerned about the hit to the stock market, which has experienced intense volatility and some of its worst trading days in years. The S&P 500 is down 10 percent since Mr. Trump’s Jan. 20 inauguration.

Speaking from the Oval Office on Wednesday, Mr. Trump said he wanted to make a deal with China. But he said what happens with his tariffs on China “depends on them.” He denied any concerns about what the tariffs are doing to small businesses, but said that the high tariff “basically means China isn’t doing any business with us.”