Early in 2017, he opened a private channel to China’s ambassador to Washington, Cui Tiankai, to try to settle Mr. Trump’s relations with the Chinese government, which had gotten off to a bumpy start after Mr. Trump spoke with Taiwan’s president. Mr. Kushner and Mr. Cui choreographed a two-day meeting that April, in which Mr. Trump played host to the Chinese leader, Xi Jinping, at Mar-a-Lago, his estate in Palm Beach, Fla.

Many saw the meeting as heralding Mr. Kushner’s role as a key player on China. But he stepped back to avoid antagonizing other senior officials, according to a White House spokesman, and amid a swirl of questions about his family’s business dealings with a Chinese conglomerate, the Anbang Group, and his sister’s offer to obtain preferential American visas for Chinese who invested in the Kushner family’s real estate projects. The spokesman, Raj Shah, said those business issues had nothing to do with Mr. Kushner’s decision.

Mr. Schulze is not the only person who has offered to act as a broker for talks between the United States and North Korea. More than a dozen people approached the State Department during the last year with claims to have connections to people high in the North Korean government, according to current and former officials. Most led nowhere, and some diplomats are doubtful that any were truly consequential.

“Over the past three administrations, the North Korean leadership has used intermediaries to try to land a summit with the American president and to bypass normal diplomatic channels,” said Michael J. Green, who worked on North Korea during the George W. Bush administration. “At other times, intermediaries who had connections to the North Korean government would make offers to help.”

Such freelance diplomacy is also not unique to North Korea. When President Barack Obama signaled an interest in talking to Iran in 2009, several people, among them a former Spanish prime minister and an Omani businessman, approached the State Department to offer to act as an intermediary. The administration later set up a secret channel to meet with Iranian officials in the Arab sultanate of Oman.

During a trip to Beijing in September, Mr. Tillerson said the administration had “a couple, three channels open to Pyongyang” that he hoped might yield a breakthrough amid escalating tensions over North Korea’s nuclear and missile programs. Mr. Trump scolded him on Twitter the next day, saying Mr. Tillerson was “wasting his time trying to negotiate with Little Rocket Man,” the derogatory nickname he used in the past for Mr. Kim.