BEIJING — The last known face-to-face contact between Kim Jong-un, the North Korean leader, and senior Chinese officials did not end well.

A member of China’s Politburo, Li Jianguo, led a small delegation to Pyongyang, North Korea’s capital, in November. He carried a letter from China’s new leader, Xi Jinping, which is said to have contained a simple message: Do not launch a ballistic missile.

Twelve days later, Mr. Kim did just that.

The relationship between North Korea and China, extolled in the past to be as close as “lips and teeth,” has faltered ever since as Mr. Kim, a political neophyte believed to be in his late 20s, has continued to defy Mr. Xi, a 59-year-old seasoned statesman.

How far the alliance between the powerhouse China and the impoverished North Korea has soured is now debated openly in the Chinese news media. Few call it a serious rift, though a spirited debate appears to be under way within the Chinese government over how to handle Mr. Kim.