Markets, which have gyrated with every twist and turn in the trade war, rose on the potential that the two sides could ultimately reach a deal. U.S. stock indexes climbed before paring back some of their gains. The S&P 500 index was up 0.29 percent at the close of the day, while the Dow Jones industrial average gained 0.17 percent.

Relations between the countries have worsened since May, when China backed away from a nearly complete deal that would have required it to codify the agreement into Chinese law, which Beijing said would infringe on its sovereignty. Since then, Mr. Trump has placed tariffs on an additional $112 billion of Chinese products and threatened further tariff increases in October and December.

China has responded to the escalation by increasing tariffs on $75 billion of American goods. Chinese state-owned companies have also suspended their purchases of American soybeans, pork and other products, a severe hit to American farmers who have already lost markets because of the trade clash.

Although Mr. Trump’s advisers publicly insist that the trade war is having no effect on the American economy, many of them are eager to calm tensions. They have been reviewing ways to avoid planned tariff increases that would result in the United States taxing nearly every Chinese toy, sneaker and computer by the end of this year.

The president pushed back on reports Thursday that he was vying for an interim deal with China that would resolve only some issues. “I’d rather get the whole deal done,” he said, before adding, “It’s something we would consider, I guess.”

The administration has been weighing whether a deal with China would be a boon or liability to the president’s re-election. His advisers have been working for months to secure an agreement strong enough to dodge criticism from both Democrats and Republicans that Mr. Trump is folding to America’s biggest economic competitor.

Some White House officials, including the president’s son-in-law, Jared Kushner, have argued that the president does not need to seal a deal with China to win over voters. Mr. Kushner and others argue that if the administration can deliver other trade successes, like passing the revised North American trade agreement and announcing a trade deal with Japan, that will be enough to help rally the base, according to people familiar with their thinking.