BEIJING—The U.S. and China asked one another to make sweeping concessions in trade talks, failing to bridge sharp divisions and raising the chances that each government will slap tariffs on tens of billions of dollars of the other country’s exports.

Top economic officials on both sides sat down for two days of talks this week after exchanging lists that contained sizeable demands. The U.S. asked China to cut its trade surplus by $200 billion, for example, while the Chinese officials sought to get Washington to ease national-security...