In a flurry of exchanges that included a recent phone call from President Obama to Mr. Xi, administration officials said, they have briefed the Chinese in detail about American plans to upgrade missile defenses and other steps to deter the increasingly belligerent threats made by North Korea’s young leader, Kim Jong-un.

China, which has been deeply suspicious of the American desire to reassert itself in Asia, has not protested publicly or privately as the United States has deployed ships and warplanes to the Korean Peninsula. That silence, American officials say, attests to both Beijing’s mounting frustration with the North and the recognition that its reflexive support for Pyongyang could strain its ties with Washington…

Chinese analysts say there are internal debates within the Communist Party and the military about how to deal with Mr. Kim, and how strongly to enforce the United Nations’ economic sanctions that China signed on to last month…

In the short run, officials said, the administration wants the Chinese to be rigorous in customs inspections to interdict the flow of banned goods to North Korea. More broadly, it wants China to persuade Mr. Kim to cease his provocations and agree to negotiations on giving up his nuclear program.